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+{"text":"\\section{\n#1}}\n\\renewcommand{\\arabic{section}.\\arabic{equation}}{\\thesection.\\arabic{equation}}\n\\begin{document}\n\\newcommand{\\ft}[2]{{\\textstyle\\frac{#1}{#2}}}\n\\newcommand{{\\hspace*{\\fill}\\rule{2mm}{2mm}\\linebreak}}{{\\hspace*{\\fill}\\rule{2mm}{2mm}\\linebreak}}\n\\newcommand{\\begin{equation}}{\\begin{equation}}\n\\newcommand{\\end{equation}}{\\end{equation}}\n\\newcommand{\\begin{eqnarray}}{\\begin{eqnarray}}\n\\newcommand{\\end{eqnarray}}{\\end{eqnarray}}\n\\newtheorem{definizione}{Definition}[section]\n\\newcommand{\\begin{definizione}}{\\begin{definizione}}\n\\newcommand{\\end{definizione}}{\\end{definizione}}\n\\newtheorem{teorema}{Theorem}[section]\n\\newcommand{\\begin{teorema}}{\\begin{teorema}}\n\\newcommand{\\end{teorema}}{\\end{teorema}}\n\\newtheorem{lemma}{Lemma}[section]\n\\newcommand{\\begin{lemma}}{\\begin{lemma}}\n\\newcommand{\\end{lemma}}{\\end{lemma}}\n\\newcommand{\\begin{array}}{\\begin{array}}\n\\newcommand{\\end{array}}{\\end{array}}\n\\newcommand{\\nonumber}{\\nonumber}\n\\newtheorem{corollario}{Corollary}[section]\n\\newcommand{\\begin{corollario}}{\\begin{corollario}}\n\\newcommand{\\end{corollario}}{\\end{corollario}}\n\\def\\twomat#1#2#3#4{\\left(\\begin{array}{cc}\n {#1}&{#2}\\\\ {#3}&{#4}\\\\\n\\end{array}\n\\right)}\n\\def\\twovec#1#2{\\left(\\begin{array}{c}\n{#1}\\\\ {#2}\\\\\n\\end{array}\n\\right)}\n\\def{\\bf Z}{{\\bf Z}}\n\\def{\\bf R}{{\\bf R}}\n\\def{\\bf C}{{\\bf C}}\n\\def{\\bf I}{{\\bf I}}\n\\def{\\bf h}{{\\bf h}}\n\\def{\\bf k}{{\\bf k}}\n\\def{\\bf g}{{\\bf g}}\n\\begin{titlepage}\n\n\\hfill\n\\vbox{\\hbox{CERN-TH\/99-53}\\hbox{hep-th\/990000}\\hbox{March, 1999}}\n\\vfill\n\\vskip 3cm\n\\begin{center}\n{\\LARGE {  On Central Charges and Hamiltonians\\\\\nfor 0-brane dynamics}$^*$}\\\\\n\\vskip 1.5cm\n  {\\bf\nR. D'Auria$^1$ S. Ferrara$^2$ and M.A. Lled\\'o$^1$\n } \\\\\n\\vskip 0.5cm\n{\\small\n$^1$ Dipartimento di Fisica, Politecnico di Torino,\\\\\n Corso Duca degli Abruzzi 24, I-10129 Torino.\\\\\nand Istituto Nazionale di Fisica Nucleare (INFN) \\\\ Sezione di Torino, Italy.\\\\\n\\vspace{6pt}\n$^2$ CERN Theoretical Division, CH 1211 Geneva 23, Switzerland.}\n\\end{center}\n\\vskip 4cm\n\n\n  {\\small\n\n\\begin{abstract}\nWe consider general properties of central charges of zero-branes and associated duality\n invariants, in view of their double role, on the bulk and on the world volume (quantum\n mechanical) theory.\n\nA detailed study of the BPS condition for the mass spectrum arising from toroidal compactifications\nis given for 1\/2, 1\/4 and 1\/8\n BPS states in any dimension. As a byproduct, we retrieve the U-duality invariant conditions\n on the charge (zero mode) spectrum and the orbit classification of BPS states preserving\n different fractions of supersymmetry.\n\nThe BPS condition for 0-branes in theories with 16 supersymmetries in\nany dimension is also  discussed.\n\\end{abstract}  }\n  \\vspace{2mm} \\vfill \\hrule width 3.cm\n{\\footnotesize\n $^*$ Supported in part by   EEC  under TMR contract\n ERBFMRX-CT96-0045,(LNF Frascati,\nPolitecnico di Torino) and by DOE grant\nDE-FGO3-91ER40662}\n \\end{titlepage}\n\\eject\n\\section{Introduction.}\n\nIn recent time, the role of duality symmetries of a dynamical theory\nencompassing quantum gravity has received increasing attention in several contexts.\n\nParticular examples where the duality takes an important role, especially in connection with\n non perturbative properties, is the AdS\/CFT correspondence \\cite{ma}, related to the horizon geometry\n of p-branes and their world-volume conformal field theory description.\n\nAnother example is the connection between M-theory compactified on a\ntorus  T$^d$ \\cite{bfss, dvv, egkr, bs, op}  and $(d+1)$\n Yang-Mills theory compactified on the dual torus $\\tilde{\\mbox{T}}^d$.\n\nMore closely related to the latter is the recent investigation of D-brane Born-Infeld\n actions and the role played by duality in explaining several properties of their\n Hamiltonian formulation and the corresponding energy spectrum of BPS\nstates \\cite{hvz}. In this\n framework it is believed that Born-Infeld non abelian gauge theories with non trivial R-R\n backgrounds, are naturally described by some generalization of gauge theories on non\n commutative tori \\cite{cds}.\n\nThe framework of non commutative geometry offers for instance,\n   a new interpretation of the\n T-duality group $O(d,d;{\\bf Z})$ of quantum mechanical systems obtained by compactifying the\n Born-Infeld action of D-branes on T$^d$. The latter occurs in type II string theory\ncompactified on\n T$^d$ \\cite{mz, bm, ks, do}.\n\nThese quantum mechanical systems have also been shown \\cite{hb}, at least for $d\\leq 4$,\n to exhibit the\n full extended U-duality symmetry\\footnote{In this paper U-duality will mean both\n the classical and quantum U-duality} E$_{d+1(d+1)}$, rather than the\nsmaller symmetry E$_{d(d)}$ present in matrix gauge theory on T$^d$,\nwhere it appears as an extension of the geometrical symmetry SL($d$) \\cite{op, egkr}.\n\nIn previous investigations, the central charge matrix $Z$  for 0-branes played a role, not\n only as central extension of supersymmetry algebra in theories with non trivial 0-brane\n background metric, but also as effective potential of the geodesic\naction of a one-dimensional Lagrangian system derived from the bulk\nEinstein-Maxwell Lagrangian, in presence of moduli fields $\\{\\phi\\}$ and\nquantized charges $q_A$ of zero-branes \\cite{gkk, fgk}.\n\nThe critical points of this potential were seen to determine the Bekenstein-Hawking\nentropy formula as the extremization of the Weinhold potential \\cite{fgk}.\n\\begin{equation}\nW={1\\over 2}\\mbox{Tr}(ZZ^\\dagger)\n\\end{equation}\nor equivalently of the BPS mass $m_{BPS}=|Z_h|$ where $|Z_h|$ is the\nhighest eigenvalue of $\\sqrt{ZZ^\\dagger}$ \\cite{fsf}.\n\nIn the world-volume description of 0-branes, the very same function $W$\nappears as Hamiltonian of the 0-brane quantum mechanics \\cite{hvz, hb, hv}\n\\begin{equation}\nH_{\\phi}(\\hat q)=\\sqrt{{1\\over N}\\mbox{Tr}(ZZ^\\dagger(\\phi,\\hat q))}, \\label{hamiltonian}\n\\end{equation}\nwhere the quantized charges are replaced by a set of Hamiltonian variables $\\hat q$,\n which belong to the same duality\nmultiplet as the quantized charges of the bulk supergravity theory\nin presence of zero-brane sources.\n\nThe appearence of the central charge in the 0-brane action in\narbitrary $D=4$ supergravity backgrounds has recently been shown to\noccur as a consequence of $\\kappa$-supersymmetry \\cite{bcfd}.\n\nIn this framework the energy spectrum of the Hamiltonian\n(\\ref{hamiltonian}) is given by the BPS mass formula of the effective\nsupergravity theory \\cite{hvz}\n\\begin{equation}\nm_{BPS}=|Z_h(\\phi, q_0)|,\n\\end{equation}\nwhere the hamiltonian variables $\\hat q$ are replaced by their zero mode part\n$q_0$ which eventually coincide with the same duality multiplet of the\nquantized charges of the bulk theory, but now with the interpretation of\n\"fluxes\" and \"momenta\"  of the world-volume hamiltonian description \\cite{op, egkr}.\n\nThese zero modes fill representations of the U-duality group $E_{d+1(d+1)}({\\bf Z})$\nfor systems with maximal supersymmetry and the BPS spectrum preserves\nsome fraction of supersymmetry depending on the particular orbit of the\ncharge vector state \\cite{fm, fg, lsp}.\n\nNote that the BPS energy $|Z_h(\\phi,q_0)|$ is not the same as replacing in\n$\\sqrt{ZZ^\\dagger(\\phi, \\hat q)}$ the zero mode $q_0$  of $q$, unless the states\nare 1\/2 BPS \\cite{hvz}, which, as we will see, can only occur if the charge duality\nmultiplet satisfies some duality invariant conditions. At the classical\nlevel, where the charges are continuous , this is equivalent to say that\nthe zero-mode part belongs to a particular orbit of the\ncharge vector representation of the duality group $G$.\n\nIt is the aim of the present investigation to derive general formulas\nof  the energy spectrum for any torus T$^d, \\; d=1,\\dots 6$ and\nprovide a new derivation of the different BPS conditions in terms of the\nU-duality invariant constraints, retrieving then the analysis of\nMaldacena and one of the authors \\cite{fm} as well as the classification of\nGunaydin and one of the authors \\cite{fg} and Lu, Pope and Stelle \\cite{lsp}.\nWe deal also with the case of 0-branes in theories in any dimension with 16\nsupersymmetries. This  is  interesting because it is related to\nheterotic strings compactified on T$^d$ or Type II theories compactified on more general\nmanifolds (such as K$_3$).\n\nThe paper is organized as follows:\n\nIn Section 3 we consider systems compactified on T$^d, \\; d=1,\\dots 4$\nwhere only 1\/2 and 1\/4 BPS states occur.\n\nIn Section 4 we consider the richer structure occurring for $d$=5,6 where\na complete understanding of the world volume theory is still missing.\n\nIn Section 5 the BPS conditions are derived for the case of theories\nwith sixteen supersymmetries in any dimension.\n\n\\section{Central charges and geometrical tools of coset spaces.}\n\nIn the present section we review the central charges for 0-branes in\ntheories with maximal supersymmetry and the BPS conditions on the\nU-multiplet of quantized charges which entail different orbits of the\n duality group which preserve different fractions of supersymmetry.\n The analysis for theories with 16 supersymmetries\nwill be considered in the last section.\n\nThe  supergravity theories describing these systems can be\nobtained in three different ways, by compactifying M theory on T$^{d+1}$\n($(d+1)$-dimensional torus)\nor type IIA and type IIB string theories on T$^{d}$. We will consider here\n supergravity theories compactified down up to $D=4$\n space-time dimensions ($d=1,\\dots 6$).\n\nSome of the results presented here overlap with previous analysis for\n$d=1,\\dots 4$, when only 1\/2 or 1\/4 BPS states are present \\cite{dvv, hvz, hv}.\nThe analysis\nof $d=5,6$ is essentially novel although the BPS conditions for 1\/2, 1\/4\nand 1\/8 BPS states were previously discussed in the literature and the\norbit classifications derived \\cite{fm, fg, lsp}.\n\n\\subsection{R-Symmetry and U-duality.}\n\nThe supersymmetry algebra of type II string theory compactified on T$^d$\ndown to $10-d$ dimensions has an R-symmetry group and a continuous\nduality group which depends on $d$. The R-symmetry is given below \\cite{cr}:\n\n\\begin{center}\n{\\bf R-symmetry group $H$}\n\\begin{eqnarray}\n&d=1  & \\mbox{U(1)}\\nonumber\\\\\n&d=2  & \\mbox{SU(2)}\\times \\mbox{U(1)}\\nonumber\\\\\n&d=3  & \\mbox{USp(4)}\\approx\\mbox{O(5)}\\nonumber\\\\\n&d=4  & \\mbox{USp(4)}\\times\\mbox{USp(4)}\\approx\\mbox{O(5)}\\times\\mbox{O(5)}\\nonumber\\\\\n&d=5  & \\mbox{USp(8)}\\nonumber\\\\\n&d=6  & \\mbox{SU(8)}\n\\end{eqnarray}\n\\end{center}\nThe U-duality groups $G$ are E$_{d+1(d+1)}$ \\cite{cr}, and the R-symmetry groups are\ntheir maximal compact subgroups. The quantum U-duality groups are  E$_{d+1(d+1)}({\\bf Z})$\n\\cite{ht}.\nBecause of the connection between M-theory and string theory, the groups\nE$_{d+1(d+1)}$ contain, both\n\\begin{equation}\n\\mbox{Gl}(d+1)\\subset  \\mbox{E}_{d+1(d+1)}\n\\end{equation}\nwhich is the classical isometry group of the moduli space of a T$^{d+1}$\ntorus in M-theory and\n\\begin{equation}\n\\mbox{O}(1,1)\\times\\mbox{O}(d,d)\\subset \\mbox{E}_{d+1(d+1)}\\quad (d\\neq 6),\\quad\n\\mbox{Sl}(2)\\times\\mbox{O}(6,6)\\subset \\mbox{E}_{7(7)} \\quad \\mbox{for}\\quad d=6,\n\\end{equation}\nwhich is the S-T duality group of string theory \\cite{wi, adf, ht}.\n\nIn string theory the $\\mbox{O}(d,d)$ group combines the geometric isometry\nof the T$^d$ torus GL$(d)$ with the shift of the antisymmetric tensor\n$B_{ij}\\mapsto B_{ij}+N_{ij}$ while the O(1,1) factor corresponds to the\ndilaton shift. The group E$_{d+1(d+1)}$ emerges from the combination of\nSl($d+1$) with O$(d,d)$ and this operation gives rise to non\nperturbative symmetries which combine the N-S-NS  and R-R  fields in\nsupermultiplets.\n\nThe spinorial charges of the supersymmetry  algebra transform  in\nrepresentations of the R-symmetry group and\nthis implies that the central charges of interest to us have\n certain symmetry and reality properties.\n\nIn the case of Lorentz scalar central charges, appropriate to 0-branes,\nthe classification goes as follows: the central charge matrix\n$Z(\\phi,q)$ is in the same representation  of the R-symmetry as  the\nvector fields $A_\\mu$ of the corresponding theory. This gives the\nfollowing result,\n\\vfill\\eject\n\\begin{center}\n{\\bf Central charge representation of the R-symmetry}\n\\begin{eqnarray}\n&d=1  &\\quad \\mbox{{\\bf 3} of O(2), ( real symmetric tensor).}\\nonumber\\\\\n&d=2  &\\quad \\mbox{{\\bf 3(+)}  of SU(2)$\\times$U(1) , (complex triplet).}\\nonumber\\\\\n&d=3  &\\quad \\mbox{{\\bf 10} of USp(4), (real antisymmetric tensor).}\\nonumber\\\\\n&d=4  &\\quad \\mbox{{\\bf 16} of USp(4)$\\times$USp(4), (bispinor (4,4)\nof O(5)$\\times $O(5)).}\\nonumber\\\\\n&d=5  &\\quad \\mbox{{\\bf 27} of USp(8), ($\\Omega$-traceless symplectic antisymmetric tensor).}\\nonumber\\\\\n&d=6  &\\quad \\mbox{{\\bf 28}  of SU(8), (complex antisymmetric tensor).}\n\\label{repre}\n\\end{eqnarray}\n\\end{center}\n\nThe previous results follow both, from a dynamical reduction of the 11\nor 10 dimensional supergravities with 32 supercharges or by an analysis\nof extended superalgebras in the appropriate dimensions \\cite{to, adf2}.\n\nSince in the original IIA theory there is only one D 0-brane (one scalar\ncentral charge) \\cite{to}, all the charges in lower dimensions come by wrapping\nbranes on the torus cycles, other than momenta and string windings.\nIn the type IIB on T$^d$, 0-branes  emerge as momenta, string windings,\nand D-branes wrapped on the torus cycles.\n\nIf we want to\ndiscuss quantum mechanical systems emerging from $d+1$ Born-Infeld\nLagrangians compactified on T$^d$, we must consider IIA D-branes\ncompactified on even dimensional tori and IIB  D-branes\ncompactified on odd dimensional tori.\n\nThe world volume description of the central charges $Z$ and quantized\ncharges $q$ is fairly well understood for the case of T$^d$ with\n$d=1,\\dots 4$. The U-duality multiplets of the quantized charges $q$\ncorrespond to fluxes, momenta, instanton number and rank of the gauge\ngroups in the world volume Yang-Mills theory. The moduli dependent\ncentral charge determines the hamiltonian  of the quantum mechanical system as well\nas the energy spectrum of the BPS states \\cite{hvz, egkr}.\n\nThe main role played by duality is that the central charge vector extends\nthe representation of the R-symmetry group to a representation of the\nfull duality group E$_{d+1(d+1)}$ acting on the vector field strength.\nThe relevant extensions are as follows \\cite{cr}\n\\begin{center}\n   {\\bf{ 0-brane representation of  U-duality group}}\n\\begin{eqnarray}\n&d=1  &\\quad \\mbox{{\\bf 2+1} of E$_2$=SL(2)$\\times$ O(1,1)}\\nonumber\\\\\n&d=2  &\\quad \\mbox{{\\bf (3,2)} of E$_3$=Sl(3)$\\times$Sl(2)}\\nonumber\\\\\n&d=3  &\\quad \\mbox{{\\bf 10} of E$_4$=Sl(5)}\\nonumber\\\\\n&d=4  & \\quad\\mbox{{\\bf 16} of E$_5$=O(5,5)}\\nonumber\\\\\n&d=5  &\\quad \\mbox{{\\bf 27} of E$_{6(6)}$}\\nonumber\\\\\n&d=6  &\\quad \\mbox{{\\bf 56} of E$_{7(7)}$}\n\\end{eqnarray}\\label{udua}\n\\end{center}\n\nThe moduli space of these theories is $G\/H$. At the string level this\nspace can be modded out further by E$_{d+1}({\\bf Z})$ \\cite{ht, wi}  something similar to\nthe fundamental domain versus the half plane for the prototype\nSl$(2,R)\/$O(2).\n\nIn Table(2.1) we present the  U-duality multiplets for 0-branes\n in the bulk and world volume description \\cite{hvz}.\n\nThe gauge fields of type II supergravity theory, as well as the Yang-Mills\nworld-volume fluxes complete U-duality multiplets of E$_{d+1(d+1)}$ for\n$d=1,\\dots 4$. These multiplets are obtained by the E$_{d(d)}$ flux and\nmomenta multiplets of matrix gauge theory on T$^d$ and by adding an\nE$_{d(d)}$ singlet, the rank of the gauge group.  For\n$d=5,6$, the Yang-Mills theory side  misses some states corresponding\nto a NS five brane and K-K monopoles \\cite{op, egkr}.\n\n\\begin{center}\n\\begin{tabular}[t]{c|c|c|}\n{\\bf d}& {\\bf Supergravity Vector Fields} & {\\bf Born-Infeld Y-M fluxes}\\\\\n\\hline\n& {\\bf IIA} \\\\\n\\hline\n2&$ Z_\\mu,g_{\\mu i},b_{\\mu i},A_{\\mu ij}$ & $\\int$Tr$F_{ij}$, $\\int$\nTr$P_i$ $\\int$Tr$E_i$, rank \\\\\n\\hline\n4& $Z_\\mu,g_{\\mu i},b_{\\mu i}, A_{\\mu ij}$ &$\\int$Tr$F_{ij}F_{kl}$,\n$\\int$Tr$P_i$, $\\int$Tr$E_i$,$\\int$Tr$F_{ij}$\\\\\n\n &$ A^D_{\\mu ijkl}$& rank\\\\\n\\hline\n 6&$Z_\\mu,g_{\\mu i},b_{\\mu i}, A_{\\mu ij}$&$\\int$Tr$F_{ij}F_{kl}F_{pq}$,\n $\\int$Tr$P_i$, $\\int$Tr$E_i$, $\\int$Tr$F_{ij}F_{kl}$\\\\\n\n & $A^D_{\\mu ijkl},Z^D_{\\mu ijklpq}$& $\\int$Tr$F_{ij}$, rank\\\\\n  &$b^{NS}_{\\mu ijklp}, g^D_{\\mu i}$& \\\\\n\\hline\n& {\\bf IIB} \\\\\n\\hline\n1& $g_{\\mu 1}, b_{\\mu 1}, b^C_{\\mu 1}$&$\\int$Tr$P$, $\\int$Tr$E$, rank\\\\\n\\hline\n3&$g_{\\mu i}, b_{\\mu i}, b^C_{\\mu i}, A_{\\mu ijk}$& $\\int$Tr$P_{i}$,\n$\\int$Tr$E_{i}$, $\\int$Tr$F_{ij}$, rank\\\\\n\\hline\n5& $g_{\\mu i}, b_{\\mu i}, b^C_{\\mu i}, A_{\\mu ijk}$&\n$\\int$Tr$P_{i}$,$\\int$Tr$E_{i}$, $\\int$Tr$F_{ij}F_{kl}$, $\\int$Tr$F_{ij}$\\\\\n\n &$b^D_{\\mu ijklp}$ & rank\\\\\n\n &$b^{NS}_{\\mu ijklp}$ & \\\\\n \\hline\n\\end{tabular}\n\\vskip 3mm\n{\\bf Table 2.1}\n\n\\end{center}\n\\vskip 5mm\n\nThe coset spaces $G\/H$ ($G\\equiv U, H\\equiv R$, in our case) can be described by choosing a representative\nin each equivalence class. If $\\phi$ denotes the coordinates of a point in $G\/H$, then the\ncoset representative will be given by an element $L(\\phi)\\in G$, in the\n equivalence class correspondig to $\\phi$, that is,  $L(\\phi)$ is a local section\nin the principal bundle $G$ over $G\/H$ with structure group $H$ . Under\nthe action of $g\\in G$  on $G\/H$ we have $\\phi\n\\mapsto \\phi_g$ and the coset representative $L(\\phi)$ will be mapped to\n$gL(\\phi)$ which is on the fiber over $\\phi_g$, so necessarily $L(\\phi_g)=\ngL(\\phi)h(\\phi)$. Taking a representation of the group $G$, (which\nis also a representation of $H$) we obtain $L(\\phi)$ as a matrix\n$L_a^\\Lambda(\\phi)$ where the indexes $a$ and $\\Lambda$ run in principle\nover the\nsame representation space, the different names being used to remind the\n covariant properties of $L$. If the representation of $G$ is reducible\nunder $H$, one can project down  a the subspace where\nthe representation of $H$ is irreducible, so the index $a$ will be\nunderstood as running\non that subspace. We will use the representations  appropriated\nto 0-brane multiplets.\n\nThe central charge is  given by\n\\begin{equation}\nZ_a(\\phi,q)= (q^TL)_a=(q^T)_\\Lambda L_a^\\Lambda(\\phi) \\label{cencha}\n\\end{equation}\n$q$ is a vector transforming under the contravariant representation of $G$,\n\\begin{equation}\nq^g=(g^{-1})^Tq\n\\end{equation}\nso the central charge is U-duality covariant in the sense that under a transformation\n\\begin{equation}\n\\phi\\mapsto \\phi_g, \\quad ({q^g}^T)_\\Lambda= (q^T)_\\Sigma (g^{-1})^\\Sigma_\\Lambda,\n\\label{trans}\n\\end{equation}\nthen $Z\\mapsto Zh$.\nIt then follows that any $H$-invariant function $I(Z)$\n is also U-duality invariant in the sense that\n\\begin{equation}\nI(\\phi_g,q^g)=I(\\phi,q), \\quad\\mbox{or}\\quad I(Zh)=I(Z) \\label{dinva}\n\\end{equation}\n\nAmong the duality invariant combinations there are some which are\n``topological'',i. e., they do not depend on the moduli \\cite{adf3, adf4}.\nThis happens when\nthe $H$-invariant is also $G$-invariant with respect to the right action of $G$. In fact,\n since $Z=q^TL$, with $L\\in G$, it is obvious that if $I(Z)$ is $G$-invariant\nfor $Z\\mapsto Zg$, then\n\\begin{equation}\nI(Zg)=I(Z)=I(q)\n\\end{equation}\nSuch objects exist for $d=5,6$ \\cite{fsf, kk}, but not for $d=1,\\dots 4$, with the implication\nthat a Bekenstein-Hawking entropy formula for 0-branes exist only in 4 and 5\ndimensions \\cite{cdcc}.\n\nWe can make a generalization of this analysis to obtain other  moduli invariant conditions.\nLet us consider now  a covariant expression,\n\\begin{equation}\nE_\\alpha(Z)=E_\\alpha(\\phi,q),\n\\end{equation}\nwhere the index  $\\alpha$ runs over the space of some representation $T$ of\n $G$ (and $H$).\nThe covariance property means that under a left $G$  transformation\n\n\\begin{equation}\nE_\\alpha(Zg)=E_\\beta(Z)T(g)^\\beta_\\alpha\n\\end{equation}\nIt follows that an equation of the form $E_\\alpha(Z)=E_\\alpha(\\phi,q)=0$ is moduli\n  independent, so $E_\\alpha(q)=0$.\n\n Now, assume that the representation $T$  admits an $H$-invariant norm\n(which is positive since $H$ is compact).\n \\begin{equation}\n\\| E_\\alpha(Z)\\|^2=g^{\\alpha\\beta} E_\\alpha E_\\beta.\n\\end{equation}\nAn equation like\n\\begin{equation}\n\\| E_\\alpha(Z)\\|^2=0 \\label{norm}\n\\end{equation}\nis in principle moduli dependent since this expression is not\n$G$-invariant. But the constraint $\\|E_\\alpha(Z)\\|=0$ implies that\n$E_\\alpha(Z)=0$, which is moduli independent so $E_\\alpha(q)=0$.\n\n\\medskip\nThe central charges satisfy some differential identities that are\ninherited from the coset representatives. To see this, let us consider\nthe algebra valued  Maurer-Cartan form in $G$, usually expressed for a\nmatrix group as\n\\begin{equation}\n\\alpha_{MC}=g(x)^{-1}dg(x)\n\\end{equation}\nwhere $x$ denotes coordinates on the group $G$. The components of the\nMaurer-Cartan form\nare left invariant forms, and one can take the pullback to $G\/H$ by the\nlocal section $L(\\phi)$, giving a local,  algebra valued left invariant form on $G\/H$\n\\begin{equation}\n\\Omega(\\phi)=L^{-1}(\\phi)dL(\\phi).\\label{mc}\n\\end{equation}\nConsider now the Cartan decomposition of the Lie algebra of $G$ as\n${\\bf g}={\\bf h}\\oplus{\\bf k}$, where ${\\bf h}$ is the Lie algebra of $H$, the\nmaximal compact subgroup of $G$, and ${\\bf k}$ can be identified with the\ntangent space at the identity coset. Since our coset spaces are\nsymmetric spaces, the following properties are satisfied,\n\\begin{equation}\n[{\\bf h},{\\bf h}]\\subset{\\bf h}, \\quad [{\\bf h},{\\bf k}]\\subset{\\bf k},\\quad\n[{\\bf k},{\\bf k}]\\subset{\\bf h}.\\label{subal}\n\\end{equation}\nThe second equation in (\\ref{subal}) means that by the adjoint, ${\\bf h}$\n acts on ${\\bf k}$ as  a representation $R$ of dimension equal to dim$(G\/H)$. We\ncan write $\\Omega$ according to this decomposition of the algebra,\n\\begin{equation}\n\\Omega=\\omega^iT_i+P^\\alpha T_\\alpha\n\\end{equation}\nwhere $\\{T_i\\}$ form a basis of ${\\bf h}$ and $\\{T_\\alpha\\}$ form a basis of ${\\bf k}$.\nThe projection of $\\Omega$  on ${\\bf h}$, $\\omega^iT_i$ is a $G$ invariant\nconnection on the  bundle with fiber ${\\bf k}$ and basis $G\/H$,\n associated to the principal bundle $G(G\/H)$ by the representation $R$ of $H$.\nWe call this bundle $E(G\/H)$. The connection is expressed in an open set as\n\\begin{equation}\n(\\omega)^\\alpha_\\beta =\\omega^iC_{i\\beta}^\\alpha\n\\end{equation}\n\nThe other components of $\\Omega$, $P^\\alpha$, provide us with the local\nexpresion of a homomorphism  between $E(G\/H)$ and the tangent bundle\n$T(G\/H)$. By means of this homomorphism we can pull back the invariant\n Riemannian connection on $G\/H$ which coincides with $\\omega$. Finally,\nthe invariant metric on $G\/H$, induced by the Cartan-Killing metric on\n$G$ can be locally represented as\n\\begin{equation}\ng_{\\mu\\nu}=\\mbox{Tr}(T_\\alpha T_\\beta)P_\\mu^\\alpha P_\\nu^\\beta.\n\\end{equation}\n(The indices ($\\mu, \\nu$) are 1-form indices on $G\/H$).\n\nWe want now to write (\\ref{mc}) using a representation of $G$ (and $H$)\nlabeled as we explained above indistinctely by indices of the type  $\\Lambda$\nor $a$, then we have\n\\begin{equation}\ndL_a^\\Lambda=L_b^\\Lambda \\omega_a^b+L_b^\\Lambda P_a^b.\n\\end{equation}\nBy defining as usual the covariant derivative with respect to $H$\n\\begin{equation}\n\\nabla_HL^\\Lambda_a=dL_a^\\Lambda-L_b^\\Lambda \\omega_a^b,\n\\end{equation}\nwe obtain\n\\begin{equation}\n\\nabla_HL^\\Lambda_a=L_b^\\Lambda P_a^b. \\label{nabla}\n\\end{equation}\nSuppose now that the representation of $H$ is reducible and we want to\nproject onto an irreducible factor. Since $\\omega_a^b$ is block diagonal,\nthe indices of type $a$ in $\\nabla_HL^\\Lambda_a$ can be understood as running on the\nsmaller representation, while $P_a^b$ will have in general off diagonal\ncomponents, so we could denote it by  $P_a^{b'}$, $b'$ running on the\nlarge representation space, but still specifying the covariant\nproperties under $H$. This happens when matter fields are  present.\n In that case the $H$ group is\na direct product $H_R\\times H_M$. $H_R$ is the R-symmetry group and\n$H_M$ is some matter flavour symmetry. We assume now (as it will happen\nin all our examples) that the representation of $G$ decomposes under $H$\nas {\\bf (1, T$_M$)}  +{\\bf (T$_R$,1)}, where {\\bf T$_M$} is a\nrepresentation of $H_M$ and   {\\bf T$_R$} is a representation of $H_R$.\nThen, there is a basis where the generic index $\\Lambda$ splits into\n$(a,I)$, where $a$ runs over the vector space representation of {\\bf T}$_R$\nand $I$ runs over the representation space of {\\bf T}$_M$. Then (\\ref{nabla})\ndecomposes as\n\\begin{eqnarray}\n\\nabla_{H_R}L^\\Lambda_a&=&L_b^\\Lambda P_a^b+L_I^\\Lambda P_a^I\\nonumber\\\\\n\\nabla_{H_M}L^\\Lambda_I&=&L_a^\\Lambda P_I^a+L_J^\\Lambda P_J^I,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\nabla_{H_R}L^\\Lambda_a&=&dL^\\Lambda_a - L^\\Lambda_b\\omega^b_a\\nonumber\\\\\n\\nabla_{H_M}L^\\Lambda_I&=&dL^\\Lambda_I - L^\\Lambda_J\\omega^J_I.\n\\end{eqnarray}\nThe central charges $Z_a=(q^TL)_a$ and matter charges $Z_I=(q^TL)_I$\nsatisfy consequently  the identities\n\\begin{eqnarray}\n\\nabla_{H_R}Z_a&=&Z_b P_a^b+Z_I P_a^I\\nonumber\\\\\n\\nabla_{H_M}Z_I&=&Z_a P_I^a+Z_J P^J_I.\n\\label{fmc}\n\\end{eqnarray}\n\nIn the forthcoming sections we will see that these properties enter in\nthe discussion of the BPS conditions and their duality invariant character.\n\n\\subsection{Orbit classification of BPS states.}\n\nIn order to further study properties of central charges which will be useful in\n the following section, we would like to remind the orbit classification of\n0-brane BPS configurations \\cite{fg, lsp}. To do so we will state some results of matrix algebra\nthat will be\nuseful for our analysis. We consider   matrices over the real,\n complex and quaternion fields, $M^r,M^c,M^q$.\nFor each of these matrices, the following polar decomposition holds\n \\begin{eqnarray}\nM^r&=&\\sqrt{M^r{M^r}^T}O\\nonumber\\\\\nM^c&=&\\sqrt{M^c{M^c}^\\dagger}U\\nonumber\\\\\nM^q&=&\\sqrt{M^q{M^q}^\\dagger}U_q\n\\end{eqnarray}\nwhere the matrices $\\sqrt{M{M}^\\dagger}$ are hermitian and  $O,U$ and $U_q$\nare orthogonal, unitary and quaternionic unitary (unitary symplectic),\n respectively.\n\nFrom this decomposition it then follows that if $M^r$ is symmetric, $M^c$ hermitian\n and $M^q$ symplectic hermitian,  they can be diagonalized by an appropriate\n transformation which is respectively orthogonal, unitary and unitary symplectic.\nInstead, for general matrices we can bring them to a diagonal form in the\nfollowing way,\n\\begin{equation}\nM_D=U_1MU^\\dagger_2\\label{diagonal}\n\\end{equation}\nwhere $U_1$ and $U_2$ belong to O(n), U(n) or USp(n) in each case.\n\nIt can also be shown that any antisymmetric matrix can be brought to a\nskew-diagonal form (normal form) by a transformation \\cite{kz}\n\\begin{equation}\nM_{SD}=UMU^T\n\\end{equation}\nwhere as before, $U$ belongs to the appropriate group.               .\n\nSince the central charge vector is a 2-tensor  representation of $H$,\nwe can always apply\n one of the above results.\n\nFrom the structure of the R-symmetry group listed above and the\n representation properties from (\\ref{repre}) we will see that\n it  follows that\nwith an R-rotation we can always diagonalize (or skew-diagonalize) the\nmatrix $Z$. For $d=1,\\dots 4$ there will be only two eigenvalues,\nthree eigenvalues for  for $d=5$ and four eigenvalues for $d=6$.\n\nThe richer structure occurring for $d=5,6$ is the  why also 1\/8 BPS\nstates occur, instead of the two possibilities of $d=1,\\dots 4$.\n\nIn the following table we list the orbits of the representations\nin Eq.(\\ref{udua}) corresponding to the 0-brane BPS configurations \\cite{fg, lsp}.\n\n\\begin{center}\n\\begin{tabular}[t]{c|c|c|c|}\n{\\bf Orbits} &{\\bf 1\/2 BPS} & {\\bf 1\/4 BPS} &{\\bf 1\/8 BPS}\\\\\n\\hline\n$d=1$ & Sl(2) or ${\\bf R}$ & Sl(2)$\\times{\\bf R} $&\\\\\n\\hline\n$d=2$ &${\\mbox{ Sl(3)}\\times\\mbox{ Sl(2)}\/ \\mbox{Gl(2)}\\propto {\\bf R}^3}$ &\n${\\mbox{Sl(3)}\\times \\mbox{Sl(2)}\/ \\mbox{Sl(2)}\\propto {\\bf R}^2}$&\\\\\n\\hline\n$d=3$ &$ {\\mbox{Sl(5)}\/(\\mbox{Sl(3)}\\times \\mbox{Sl(2)})\\propto {\\bf R}^6}$ &\n$ {\\mbox{Sl(5)}\/\\mbox{O(2,3)}\\propto {\\bf R}^4}$&\\\\\n\\hline\n$d=4$& ${\\mbox{O(5,5)}\/\\mbox{Sl(5)}\\propto{\\bf R}^{10}}$ &\n${\\mbox{O(5,5)}\/\\mbox{O(3,4)}\\propto {\\bf R}^8}$&\\\\\n\\hline\n$d=5$ & ${\\mbox{E}_{6(6)}\/\\mbox{O(5,5)}\\propto{\\bf R}^{16}}$ &\n${\\mbox{E}_{6(6)}\/\\mbox{O(4,5)}\\propto{\\bf R}^{16}}$ &$\n{\\mbox{E}_{6(6)}\/\\mbox{F}_{4(4)}}$\\\\\n\\hline\n$d=6$ & ${\\mbox{E}_{7(7)}\/\\mbox{E}_{6(6)}\\propto{\\bf R}^{27}}$ &\n${\\mbox{E}_{7(7)}\/(\\mbox{O(5,6)}\\propto{\\bf R}^{32})\\times{\\bf R}}$ &\n${\\mbox{E}_{7(7)}\/\\mbox{F}_{4(4)}\\propto{\\bf R}^{26}},$\\\\\n&&&$ {\\mbox{E}_{7(7)}\/\\mbox{E}_{6(2)}}$\\\\\n \\hline\n\\end{tabular}\n\\vskip 3mm\n{\\bf Table 2.2}\n\\end{center}\n\\vskip 5mm\nThese orbits correspond, for $d=1,\\dots 4$ to the possibility of having\n two coinciding eigenvalues (1\/2 BPS) or not (1\/4 BPS) for the central\n charge matrix. For $d=5,6$, 1\/2 BPS correspond to 3 and 4 coinciding\n eigenvalues respectively, 1\/4 BPS orbits correspond to 2 equal eigenvalues\n and 2 pairs of equal eigenvalues respectively and 1\/8 BPS orbits\n correspond to all different eigenvalues. In $d=6$ there are there two kinds\nof 1\/8 BPS orbits depending whether the quartic invariant vanishes  or not\n(light-like or time-like orbit)\\cite{fm, fg}.\n\nIn the following section we will see that, in spite of the fact that such\nstatements look moduli dependent, they are actually moduli\nindependent and therefore U-duality invariant, as expected from physical\nconsiderations.\n\n\\section{BPS spectrum for 0-branes in type II string theory compactified\non T$^d$, $d=1,\\dots 4$}\n\nIn the present section we consider the BPS spectrum and the central\ncharge matrix  for 0-branes in the cases when only 1\/2 and 1\/4 BPS\nstates exist. These are the cases where the central charge has only two\neigenvalues, as it happens for $d=1,\\dots 4$.\n\nLet us consider the relevant anticommutators, containing the scalar\ncentral charge,\n\\begin{eqnarray}\nd=1\\quad &\\{Q_i,Q_j\\}&=Z_{ij}\\quad (i,j=1,2);\\; Z_{ij}\\mbox{ real symmetric}\\nonumber\\\\\nd=2\\quad &\\{Q_A,Q_B\\}&=Z_I\\sigma_{AB}^I=Z_{AB}\\quad ( A,B=1,2);\\nonumber\\\\\n&&Z^I\\;\\mbox{complex},\\quad Z_{AB}\\; \\mbox{symmetric},  \\nonumber\\\\\nd=3\\quad &\\{Q_a,Q_b\\}&=Z_{IJ}(\\gamma^{IJ})_{ab}=Z_{ab}\\quad\n(a,b=1,\\dots 4);\\nonumber\\\\\n&& Z_{ab}\\; \\mbox{ symmetric and symplectic} \\nonumber\\\\\nd=4\\quad &\\{Q_a,Q_b'\\}&=Z_{ab'}\\quad\n(a,b'=1,\\dots 4);\\; Z_{ab'}\\; \\mbox{symplectic}\n\\end{eqnarray}\nwhere $\\sigma^I$ are the Pauli matrices and  $\\gamma^{IJ} = 1\/2 [\\gamma^I\n, \\gamma^J ]$ ($ \\gamma^I$ are the O(5) gamma matrices).\nWe say that  $Z_{ab}$ is a symplectic (or quaternionic) matrix if:\n\\begin{equation}\n\\bar Z =- \\Omega Z \\Omega \\label{sym}\n\\end{equation}\nwhere $\\Omega$ is the bilinear form invariant under\nUSp(4) which satisfies\n\\begin{equation}\n\\Omega = \\bar \\Omega = -  \\Omega^T = - \\Omega^{-1}\n\\end{equation}\nThe indices $a,b$ in the gamma matrices are raised and lowered with $\\Omega$.\n\n\\paragraph{ Cases $d=1,2$.} In these cases the chare matrix $Z$ is\n2$\\times$2 and has two independent eigenvalues. The 1\/2 BPS conditions\ncorrespond to these eigenvalues equal in magnitude.  We consider separately\nboth cases.\n\n\\medskip\n\n$\\bullet$ For $d=1$, $Z$ is real and symmetric. We can decompose it as\n\\begin{equation}\nZ_{ij}=Y\\delta_{ij} +Z^\\alpha (T_\\alpha)_{ij}\n\\end{equation}\nwhere $\\alpha=1,2$, $T_1=\\sigma_1$ and $T_2=\\sigma_3$ (the Pauli\nmatrices).\nThe characteristic equation for $Z$ is\n\\begin{equation}\n\\lambda^2 - \\mbox{Tr} Z \\, \\lambda +\\mbox{det}Z =0 \\label{simple}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n&& \\mbox{Tr} Z = 2 Y \\\\\n&&\\mbox{det}Z = Y^2 - Z^{\\alpha} Z_{\\alpha}\n\\end{eqnarray}\nIt then follows that  the two solutions of (\\ref{simple}) satisfy\n $|\\lambda_1|=|\\lambda_2|$ if either\n\\begin{equation}\n\\mbox{Tr} Z=0\n\\end{equation}\nor\n\\begin{equation}\n\\mbox{det} Z =1\/4 (\\mbox{Tr}Z)^2\n\\end{equation}\nthen implying\n\\begin{equation}\nYZ^{\\alpha}Z_{\\alpha} =0.\\label{discri}\n\\end{equation}\n\nIt is obvious that  condition (\\ref{discri}) is only O(2) or R-invariant,\nbut\nthe unique solution  $YZ_\\alpha=0$, is an SL(2)$\\times$O(1,1) or U-invariant. This is\nthe first example of a condition of the type (\\ref{norm}). So we\nretrieve the result of Ref.\\cite{fm} for $d=1$, \\,1\/2 BPS states\n\\begin{equation}\nq=0 \\quad \\mbox {or}\\quad q_\\alpha=0\\quad\n\\end{equation}\n\\medskip\n\n$\\bullet$ In the $d=2$ case the hermiticity condition is lacking,\nbut still the matrix $Z^I\\sigma_I$ can be diagonalized with real\neigenvalues using (\\ref{diagonal}), where the difference between $U_1$\nand $U_2$ is simply a phase.  We  just use  a transformation\nof the R-symmetry group U(2), with SU(2) acting on the $\\sigma$-matrices\nand U(1) acting  as a phase on $Z^I$.\n\nIn fact, note that\n\\begin{equation}\nZ_IZ^I=(A_I+iB_I)(A^I+iB^I)=A_IA^I -B_IB^I+2iA_IB^I\n\\end{equation}\nTherefore, with a U(1) transformation we bring  $A_IB^I$ to zero, which\nmeans that $\\vec{A}$ and $\\vec{B}$ are orthogonal vectors, so by an\northogonal transformation we can bring them to coincide with the axes,\nand only two real numbers (related to the two eigenvalues) are left.\n\nWe proceed by diagonalizing the hermitian matrix $ZZ^\\dagger$. The\nsquare root of the eigenvalues will be the eigenvalues of $Z$, that we\ndenote by $\\lambda_1, \\lambda_2$. (In this way we include both cases, when\nthe eigenvalues are equal and when they are opposite in sign).We have :\n\\begin{equation}\n\\mbox{Tr}ZZ^\\dagger=\\lambda_1^2+\\lambda_2^2, \\quad \\mbox{Tr}(ZZ^\\dagger)^2\n=\\lambda_1^4 +\\lambda_2^4,\\quad \\mbox{det}ZZ^\\dagger=\\lambda_1^2\\lambda_2^2\n\\end{equation}\nwhere $\\lambda_i$ are the eigenvalues obtained  as in (\\ref{diagonal}).\nFrom the characteristic equation for $ZZ^\\dagger$ we have\n\\begin{equation}\n\\lambda_{1,2}^2={1\\over 2}[\\mbox{Tr}ZZ^\\dagger\\pm\\sqrt{2\\mbox{Tr}(ZZ^\\dagger)^2\n-(\\mbox{Tr}ZZ^\\dagger)^2}] \\label{eigen}\n\\end{equation}\nUsing now the properties of the $\\sigma$ matrices,\n\\begin{equation}\nZZ^\\dagger=Z^I\\sigma_I {\\bar Z^J}\\sigma_J=Z^I{\\bar Z_I}{\\bf I} +i\\epsilon_{IJK}\n Z^I{\\bar Z^J}\\sigma^K\n\\end{equation}\nWe denote $\\hat Z_K=i\\epsilon_{IJK}Z^I{\\bar Z^J}$. Then we have\n\\begin{equation}\n\\mbox{Tr}ZZ^\\dagger=2Z^I{\\bar Z_I},\\quad \\mbox{Tr}(ZZ^\\dagger)^2\n=2[(Z^I{\\bar Z_I})^2+\\hat{Z}_I\\hat{Z}_I],\n\\end{equation}\nHence, the discriminant in (\\ref{eigen}) is given by\n\\begin{equation}\n\\mbox{Tr}(ZZ^\\dagger )^2-{1\\over 2}(\\mbox{Tr}ZZ^\\dagger)^2=2\\hat\nZ^I\\hat Z_I.\n\\end{equation}\nWe set $Z^I_1=A^I,\\; Z^I_2=B^I$. The 1\/2 BPS condition\n $\\hat{Z}_I\\hat{Z}^I=0$, can be written as\n\\begin{equation}\n\\|\\epsilon^{\\alpha \\beta}Z_\\alpha^IZ^J_\\beta\n\\epsilon_{KIJ}\\|=0\\Rightarrow \\epsilon^{\\alpha \\beta}Z_\\alpha^IZ^J_\\beta\n\\epsilon_{KIJ}=0.\\label{epsilon}\n\\end{equation}\nwhere $\\|\\;\\|$ is the O(3)$ \\times $O(2) invariant norm. As before,\n the condition obtained is  actually invariant under SL(3)$\\times $SL(2), so we obtain\nthe  moduli independent condition of Ref.\\cite{fm}:\n\\begin{equation}\n\\epsilon^{\\alpha \\beta}q_\\alpha^Iq^J_\\beta \\epsilon_{KIJ}=0.\n\\end{equation}\n\n\\paragraph{Cases  $d=3,4$.}\n\nIn these cases the matrix $Z$ is 4-dimensional, but because of its symplectic\nproperty (\\ref{sym})\nthere are only two independent eigenvalues (two pairs of equal\neigenvalues), $\\lambda_{1,2}$.\nIndeed, we find\n\\begin{eqnarray}\n&&\\mbox{Tr}ZZ^\\dagger=2(\\lambda_1^2+\\lambda_2^2)\\nonumber\\\\\n&&\\mbox{Tr}(ZZ^\\dagger)^2=2(\\lambda_1^4+\\lambda_2^4).\n\\end{eqnarray}\nThe characteristic equation  (or better, its square root) is\n\\begin{equation}\n\\lambda^2 -{1\\over 2}\\mbox{Tr}ZZ^\\dagger\\lambda +(\\mbox{det}ZZ^\\dagger)^{1\/2}=0,\n\\end{equation}\nwith\n\\begin{equation}\n(\\mbox{det}ZZ^\\dagger)^{1\/2}={1\\over 8}(\\mbox{Tr}ZZ^\\dagger)^2-{1\\over 4}\n\\mbox{Tr}(ZZ^\\dagger)^2.\n\\end{equation}\nThe roots are\n\\begin{equation}\n\\lambda_{1,2}^2={1\\over 2}\\left({1\\over 2}\\mbox{Tr}ZZ^\\dagger\\pm \\sqrt{\\mbox{Tr}(ZZ^\\dagger)^2\n-{1\\over 4}(\\mbox{Tr}ZZ^\\dagger)^2}\\right)\n\\end{equation}\n\nWe consider now the two cases separately,\n\\medskip\n\n$\\bullet$ In the $d=3$ case we can switch from Sp(4) to O(5) indices by\nsetting:\n\\begin{equation}\nZ_{ab}=Z_{IJ}(\\gamma^{IJ})_{ab},\\quad (I,J=1,\\dots 5)\n\\end{equation}\nwhere\n$Z_{IJ}$ is real and antisymmetric and\n\\begin{equation}\n\\gamma^{IJ}={1\\over 2}[\\gamma^I,\\gamma^J].\n\\end{equation}\nIt is clear that $Z_{IJ}$ can be skew-diagonalized with an O(5)\ntransformation, so $Z_{ab}$ can be diagonalized with a USp(4) transformation.\nFrom the relation\n\\begin{equation}\n{1\\over 2}\\{ \\gamma^{IJ},\\gamma^{KL}\\}=(g^{IJ}g^{KL}- g^{JK}g^{IL}+\\epsilon^{IJKLP}\\gamma_P )\n\\end{equation}\nit follows that\n\\begin{equation}\nZZ^\\dagger=Z^2{\\bf I} + Z^P\\gamma_P,\n\\end{equation}\nwhere\n\\begin{equation}\nZ^2=2Z^{PQ}Z_{PQ},\\quad Z^P=\\epsilon^{PIJKL}Z_{IJ}Z_{KL}.\n\\end{equation}\nFrom this, we have\n\\begin{eqnarray}\n\\mbox{Tr}ZZ^\\dagger&=&4Z^2\\nonumber\\\\\n\\mbox{Tr}(ZZ^\\dagger)^2&=&4Z^4+4Z^PZ_P.\n\\end{eqnarray}\nThe 1\/2 BPS condition becomes:\n\\begin{equation}\n\\mbox{Tr}(ZZ^\\dagger)^2 -{1\\over 4}(\\mbox{Tr}ZZ^\\dagger)^2=4Z^PZ_P=0\n\\end{equation}\nwhich  is o(5) invariant and implies\n\\begin{equation}\nZ^P=\\epsilon^{PIJKL}Z_{IJ}Z_{KL}=0 \\label{epsilon3}\n\\end{equation}\n\nEquation(\\ref{epsilon3}) is SL(5) invariant when $Z^{IJ}$ is in the 10-dimensional\n representation of SL(5), and therefore it is moduli independent, giving\n the result of Ref. \\cite{fm},\n\\begin{equation}\n\\epsilon^{PIJKL}q_{IJ}q_{KL}=0\n\\end{equation}\n\n\\medskip\n\n$\\bullet$ The $d=4$ case was already discussed in Ref.\\cite{dvv, hvz}, but we outline it  here for\ncompleteness. In this case, the matrix $Z_{ab'}$ is a general O(5) bispinor.\nHowever, since its square is hermitian it decomposes as\n\\begin{eqnarray}\nZZ^\\dagger&=&Z^2{\\bf I} +{Z_{(l)}}^P\\gamma_P,\\quad p=1,\\dots 5\\nonumber\\\\\nZ^\\dagger Z&=& Z^2{\\bf I} +{Z_{(r)}}^P\\gamma_P\n\\quad \\mbox{with} \\quad\n {Z_{(l)}}^P{Z_{(l)}}_P=  {Z_{(r)}}^P{Z_{(r)}}_P.\n\\end{eqnarray}\nwhere the subindices $l,r$ refer to the two $O(5)$ factors of the\n$R-$symmetry group.\nIt follows that\n\\begin{equation}\n\\mbox{Tr}ZZ^\\dagger=4Z^2,\\quad\n\\mbox{Tr}(ZZ^\\dagger)^2 =4Z^4+4 {Z_{(l)}}^P{Z_{(l)}}_P.\n\\end{equation}\nThe 1\/2 BPS condition is then :\n\\begin{equation}\n\\mbox{Tr}(ZZ^\\dagger)^2 -{1\/4}(\\mbox{Tr}ZZ^\\dagger)^2=4{Z_{(l)}}^P{Z_{(l)}}_P=0.\n\\end{equation}\nThe equations ${Z_{(l)}}^P{Z_{(l)}}_P= {Z_{(r)}}^P{Z_{(r)}}_P=0$ imply that the O(5)\nvectors $Z^P_{(r)}$, $Z^P_{(l)}$ vanish. This is an O(5,5) invariant statement. $(Z,Z^\\dagger)$\nform the 16 dimensional (chiral spinor) representation of O(5,5) and the O(5,5)\n10 dimensional (light-like vector) $(\\mbox{Tr} \\gamma ZZ^\\dagger,\n\\mbox{Tr} \\gamma Z^\\dagger Z)$ then vanishes when $|Z_{(l)}|=|Z_{(r)}|=0$.\nWe then retrieve the condition of Ref.\\cite{fm} on the quantized charges in the spinor\nrepresentation of O(5,5).\n\n\\section{BPS spectrum for the $d\\!=\\!5,6$ dimensional cases.}\n\nIn this section we will examine the more interesting cases of $d=5,6$,\ncorresponding to supergravity compactified down to $D=4,5$ dimensions\nrespectively.\n\nThe different BPS states, preserving some fraction of supersymmetry,\nare classified by the orbits of E$_{6(6)}$ and E$_{7(7)}$\nrespectively as given in Table 2.2\n\nTo put this analogy in perspective it is useful to parametrize the set of BPS\n charges  allowed by the duality constraints by their eigenvalues and some\n angular variables (which can be removed by an R-symmetry transformation \\cite{fg}). The\n duality constraints which follow from the BPS conditions are precisely those\n constraints which do not depend on these extra angular variables and which can\nbe removed by an $H$ transformation in $G$. These constraints will give\ndifferent orbits corresponding to different BPS conditions on the 0-brane\n charges.\n\n\\begin{center}\n{\\bf Orbits of the BPS energy levels for $d=5,6$}\n\\bigskip\n\\begin{tabular}[t]{c|c|c|c|c|}\n& Orbit & dim.& eigenv. & angles\\\\\n\\hline\n d=5   \\\\\n\\hline\n1\/2 BPS&E$_{6(6)}$\/O(5,5)$\\propto{\\bf R}^{16}$ & 17&1&16=dim(USp(8)\/O(5)$\\times$O(5)) \\\\\n\\hline\n1\/4 BPS&E$_{6(6)}$\/O(4,5)$\\propto{\\bf R}^{16}$ & 26&2&24=dim(USp(8)\/O(4)$\\times$O(4)) \\\\\n\\hline\n1\/8 BPS&E$_{6(6)}$\/F$_{4(4)}$ & 26+1&3&24=dim(USp(8)\/USp(2)$^4$) \\\\\n\\hline\n d=6   \\\\\n \\hline\n 1\/2 BPS&E$_{7(7)}$\/E$_{6(6)}\\propto{\\bf R}^{27}$ & 28&1&27=dim(SU(8)\/USp(8)) \\\\\n \\hline\n1\/4 BPS&E$_{7(7)}$\/(O(5,6)$\\propto{\\bf R}^{32})\\times{\\bf R}$ & 45&2&43=dim(SU(8)\/USp(4)$^2$) \\\\\n\\hline\n1\/8 BPS&E$_{7(7)}$\/F$_{4(4)}\\propto{\\bf R}^{26}$ & 55&4&51=dim(SU(8)\/USp(2)$^4$) \\\\\n&E$_{7(7)}$\/E$_{6(2)}$& 55+1&5&51=dim(SU(8)\/USp(2)$^4$) \\\\\n \\hline\n\\end{tabular}\n\\vskip 3mm\n{\\bf Table 4.1}\n\n\\end{center}\n\\vskip 5mm\n\nThe different orbits of different BPS levels will correspond to different\n solutions of the characteristic equation of the central charge matrix (or\n its square). These different solutions will be characterized by invariant\n constraints which are moduli independent in spite of the fact that the\n eigenvalues of the matrix are moduli dependent. Becuse of this, the orbits\n are simply given by invariant constraints on the ``quantized'' charges, as found\n in Ref.\\cite{fm}.\n\n\\paragraph{Case $d=6$.}\n\nWe consider the E$_7$ quartic invariant \\cite{cr, adf4, kk}\n\\begin{equation}\nI=4\\mbox{Tr}(Z\\bar Z)^2 -(\\mbox{Tr}Z\\bar Z)^2+2^4({\\it Pf}Z+{\\it Pf}\\bar Z)\n \\label{quartic}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\it Pf}Z={1\\over 2^44!}\\epsilon^{ABCDRPGH}Z_{AB}Z_{CD}Z_{RP}Z_{GH}.\n\\end{equation}\nWe want to consider second derivatives of the above quartic invariant\nthat could give us covariant equations. The antisymmetric matrix\n$Z_{AB}$ is in the 28-dimensional representation of SU(8), while we can\nexpress symbolically $Z_{\\bf 56}=(Z_{AB},\\bar Z^{AB})$, in the 56-dimensional\nrepresentation of E$_7$ (${\\bf 56=28 +\\bar{28}}$). Taking the  the second derivative\n\\begin{equation}\n{\\partial^2 I\\over \\partial Z_{\\bf 56}\\partial Z_{\\bf 56}}\\left|_{Adj_{E_7}}\\right.,\\label{cova}\n\\end{equation}\nwill give us a quadratic polynomial which is a symmetric tensor, in the\n$({\\bf 56}\\times {\\bf 56})_S={\\bf 1596}$\nrepresentation of E$_7$ which is not irreducible and decomposes as {\\bf 1463\n+ 133}. {\\bf 133} is the Adj$_{E_7}$, so we can project on that\n space as indicated above (\\ref{cova}).\nSince {\\bf  133} decomposes as {\\bf 63+70} under SU(8), the expression\n(\\ref{cova})  splits into the two following SU(8) covariant\npolynomials\n\\begin{equation}\n{\\partial^2 I\\over \\partial Z_{AB}\\bar \\partial Z^{CB}}\\left|_{Adj_{SU(8)}}\\right.\\approx\n( Z_{AB}\\bar  Z^{CB}-{1\\over 8}\\delta_A^C Z_{PQ}\\bar\nZ^{PQ})=V_A^C.\\label{vac}\n\\end{equation}\n\\begin{equation}\n{\\partial^2 I\\over \\partial Z_{[AB}\\partial Z_{CD]}}-{1\\over 4!}\\epsilon^{ABCDPQRS}\n{\\partial^2 I\\over \\partial \\bar Z^{[AB}\\partial \\bar Z^{CD]}}=V^+_{[ABCD]}.\n\\label{selfdual}\n\\end{equation}\nThe 1\/2 BPS condition is the E$_7$ invariant statement $ V_A^C=0$\nand $V^+_{[ABCD]}=0$. This is the constraint\n imposed in Ref.\\cite{fm} on the quantized\n56 electric and magnetic charges defining a 1\/2 BPS configuration.\nThe equation $ V_A^C=0$ implies that the matrix $ZZ^\\dagger$ has four\ncoinciding eigenvalues (that is, it is a multiple of the identity),\nwhile  the equation $V^+_{[ABCD]}=0$ implies\nthat the eigenvalues of $Z$ are real.\n\nThe vanishing of (\\ref{selfdual}) follows from the vanishing of\n(\\ref{vac}) and the differential relations (\\ref{fmc}) satisfied by $Z_{AB}$\n\\cite{adf3}, which in this case take the form\n\\begin{equation}\n\\nabla_{SU(8)}Z_{AB}={1\\over 2}\\epsilon_{ABCD}\\bar Z^{CD}.\n\\end{equation}\n\nWe now want to consider more general cases. The\ncharacteristic equation (or better,  its square root) is given by\n\\begin{equation}\n\\sqrt{\\mbox{det}(ZZ^\\dagger-\\lambda{\\bf I})} =\\prod_{i=1}^4(\\lambda-\\lambda_i)=\n\\lambda^4+a\\lambda^3+b\\lambda^2+c\\lambda+d=0\n\\end{equation}\nwhere\n\\begin{eqnarray}\na&=&-(\\lambda_1+\\lambda_2+ \\lambda_3+ \\lambda_4)\\nonumber\\\\\n&=&-{1\\over2}\\mbox{Tr}ZZ^\\dagger\\nonumber\\\\\nb&=&\\lambda_1\\lambda_2+ \\lambda_1\\lambda_3+ \\lambda_1\\lambda_4+\n\\lambda_2\\lambda_3+ \\lambda_2\\lambda_4+ \\lambda_3\\lambda_4\\nonumber\\\\\n&=&{1\\over 4}[{1\\over 2}(\\mbox{Tr}ZZ^\\dagger)^2- \\mbox{Tr}(ZZ^\\dagger)^2]\n\\nonumber\\\\\nc&=&-(\\lambda_1\\lambda_2\\lambda_3+\\lambda_1\\lambda_2\\lambda_4+\\lambda_1\n\\lambda_3\\lambda_4+\\lambda_2\\lambda_3\\lambda_4)\\nonumber\\\\\n&=&-{1\\over 6}\\left({1\\over 8}(\\mbox {Tr}ZZ^\\dagger)^3 + \\mbox{Tr}(ZZ^\\dagger)^3-\n{3\\over 4} \\mbox{Tr}ZZ^\\dagger \\mbox{Tr}(ZZ^\\dagger)^2\\right)\\nonumber\\\\\nd&=&\\lambda_1\\lambda_2\\lambda_3\\lambda_4\\nonumber\\\\\n&=&{1\\over 4}\\left({1\\over 96} (\\mbox{Tr}ZZ^\\dagger)^4 +{1\\over 8}( \\mbox{Tr}\n(ZZ^\\dagger)^2)^2+{1\\over 3} \\mbox{Tr}(ZZ^\\dagger)^3 \\mbox{Tr}ZZ^\\dagger\\right.\n\\nonumber \\\\\n&&\\left. -{1\\over 2} \\mbox{Tr}(ZZ^\\dagger)^4-{1\\over 8} (\\mbox{Tr}ZZ^\\dagger)^2\n \\mbox{Tr}(ZZ^\\dagger)^2\\right)\\label{abcd}\n\\end{eqnarray}\n\n In the case of two pairs of equal roots we have\n\\begin{equation}\n\\prod_{i=1}^4(\\lambda-\\lambda_i)=(\\lambda-\\lambda_1)^2(\\lambda-\\lambda_2)^2.\n\\end{equation}\n This implies the following relations among the coefficients\n \\begin{eqnarray}\n c&=&{1\\over 2}a(b-{1\\over 4}a^2)\\nonumber\\\\\n d&=&{1\\over 4}(b-{1\\over 4}a^2)^2 \\label{conditions}\n \\end{eqnarray}\nwhich imply the following relations among the invariants,\n\\begin{eqnarray}\n&&{32\\over 3}\\mbox{Tr}(ZZ^\\dagger)^3=4\\mbox{Tr}ZZ^\\dagger\\mbox{Tr}(ZZ^\\dagger)^2 -\n{1\\over 3}(\\mbox{Tr}ZZ^\\dagger)^3\\nonumber\\\\\n&&(\\mbox{det}ZZ^\\dagger)^{1\/2}={1\\over 64}[\\mbox{Tr}(ZZ^\\dagger)^2-\n{1\\over 4}(\\mbox{Tr}ZZ^\\dagger)^2]^2.\\label{det}\n\\end{eqnarray}\nThe eigenvalues are given by the expression\n\\begin{equation}\n\\lambda_{1,2}={1\\over 8}\\mbox{Tr}ZZ^\\dagger\\pm{1\\over 2}\\sqrt{{1\\over 2}\n\\mbox{Tr}(ZZ^\\dagger)^2-{1\\over 16}(\\mbox{Tr}ZZ^\\dagger)^2}\n\\end{equation}\nbeing the BPS mass, $m_{BPS}^2$ the highest eigenvalue (+ sign).\n\nWe want now to show how the 1\/4 BPS condition follows from the\nE$_7$ invariance. Let us consider the E$_7$ covariant constraint\n\\begin{equation}\n {\\partial I\\over \\partial Z_{AB}}=0\\quad(\\Rightarrow\n {\\partial I\\over \\partial\\bar Z^{AB}}=0)\\label{inv1}.\n \\end{equation}\n where $I$ is the invariant from (\\ref{quartic}). From this, the\nfollowing quartic SU(8) invariant equations follow,\n\\begin{eqnarray}\n {\\partial I\\over \\partial Z_{AB}}Z_{AB}+{\\partial I\\over \\partial\n\\bar Z^{AB}} \\bar Z^{AB}&=&4I=0\\label{first}\\\\\n {\\partial I\\over \\partial Z_{AB}}Z_{AB}-{\\partial I\\over \\partial\n \\bar Z^{AB}} \\bar Z^{AB}&=&0.\\label{second}\n\\end{eqnarray}\nThe second equation implies that the Pfaffian of $Z$ is real, so\n\\begin{equation}\n{\\it Pf}Z={\\it Pf} Z^\\dagger\n\\end{equation}\nand therefore\n\\begin{equation}\n({\\it Pf}Z)^2=(\\mbox{det}ZZ^\\dagger)^{1\/2}.\\label{pfaf}\n\\end{equation}\nPlugging (\\ref{pfaf}) into (\\ref{first}) and squaring, it\n gives $(\\mbox{det}ZZ^\\dagger)^{1\/2}$ as in (\\ref{det}).\n\nIn the same way one can show that the  equation giving $\\mbox{Tr}(ZZ^\\dagger)^3$ as in (\\ref{det}) is\nthe SU(8) invariant equation\n\\begin{equation}\n {\\partial I\\over \\partial Z_{AB}}{\\partial I\\over \\partial \\bar Z^{AB}}=0\\label{inv2}.\n \\end{equation}\n\nIn the generic case the 1\/8 BPS states will correspond to 4 different\neigenvalues. They are explicitely given as follows. Define the quantities\n\\begin{eqnarray}\nu&=& b^2 +12d-3ca\\nonumber\\\\\nv&=& 2b^3+27c^2-72bd -9abc +27 da^2\\nonumber\\\\\nw&=&\\left({v+\\sqrt{v^2-4u^3}\\over 2}\\right)^{1\/3}\\nonumber\\\\\ns&=&\\sqrt{{a^2\\over 4}-{2b\\over 3}+{u\\over 3w}+{w\\over 3}}\n\\end{eqnarray}\nThen,\n\\begin{eqnarray}\n\\lambda_{1,2}=-{a\\over 4}+{s\\over 2}\\pm{1\\over 2}\\sqrt{{a^2\\over 2}-{4b\\over 3}\n-{a^3-4ab +8c\\over 4s}-{u\\over 3w}-{w\\over 3}}\\nonumber\\\\\n\\lambda_{3,4}=-{a\\over 4}-{s\\over 2}\\pm{1\\over 2}\\sqrt{{a^2\\over 2}-{4b\\over 3}\n+{a^3-4ab +8c\\over 4s}-{u\\over 3w}-{w\\over 3}}.\n\\end{eqnarray}\nThe BPS mass, $m_{BPS}^2$ is $\\lambda_1$. This is the actual\ndetemination of the energy spectrum for 1\/8 BPS states in terms of the\nduality invariant quantities (\\ref{abcd}).\n\nIt is amusing that analytic expressions for the roots of a polynomial\nexist only up to quartic equations, as found by Galois \\cite{ed}, and\nthis is precisely what is required by maximal supersymmetry ($N=8$ at $D=5,6$).\n\n\\paragraph{Case $d=5$.}\n\nThe   central charge $\\hat Z_{AB}$ is a symplectic, $\\Omega$-traceless antisymmetric  matrix;\n\\begin{equation}\n \\bar {\\hat Z}=-\\Omega\\hat Z\\Omega,\\quad \\hat Z^T=-\\hat Z,\\quad \\mbox{Tr}\\hat Z\\Omega=0.\n\\end{equation}\nThis implies that the matrix\n\\begin{equation}\nZ=\\hat Z \\Omega\n\\end{equation}\nis hermitian traceless. The characteristic equation for $Z$ becomes\n\\begin{equation}\n\\sqrt{\\mbox{det}Z-\\lambda {\\bf I}}=\\prod_{i=1}^4(\\lambda-\\lambda_i)\n=\\lambda^4+b\\lambda^2 +c\\lambda +d=0\n\\end{equation}\nwhere\n\\begin{eqnarray}\nb&=& -{1\\over 4}\\mbox{Tr}Z^2\\nonumber\\\\\nc&=&  -{1\\over 6}\\mbox{Tr}Z^3\\nonumber\\\\\nd&=&  {1\\over 8}\\left({1\\over 4}(\\mbox{Tr}Z^2)^2-\\mbox{Tr}Z^4\\right)\n\\end{eqnarray}\nA 1\/4 BPS state is a state for which $c=0$. This is an E$_6$ invariant\nstatement since $c=I_3$ is the E$_6$ cubic invariant. In this case we get\n\\begin{equation}\n2\\lambda_{1,2}^2= {1\\over 4}\\mbox{Tr}Z^2\\pm \\sqrt{ {1\\over 2}\\mbox{Tr}Z^4\n -{1\\over 16}(\\mbox{Tr}Z^2)^2}\n \\end{equation}\n The discriminant is related to the modulus of the USp(8) (and E$_{6(6)}$) vector\n \\begin{equation}\n V_B^A={\\partial I\\over \\partial Z_A^B} \\approx Z^C_AZ^B_C-{1\\over 8}Z^C_DZ^D_C\\delta_A^B\n \\end{equation}\n Indeed,\n \\begin{equation}\n \\mbox{Tr}V^2= \\mbox{Tr}Z^4-{1\\over 8} (\\mbox{Tr}Z^2)^2.\n \\end{equation}\n The condition for 1\/2 BPS is that the discriminant vanishes. Therefore,\nthis implies, by positivity, $V=0$, which is an E$_6$ invariant statement\n\\begin{equation}\n{\\partial I\\over \\partial Z_A^B }=0\n\\end{equation}\nWe therefore have retrieved the results of  Maldacena and one of the\nauthors \\cite{fm}.\n\nFor the 1\/8 BPS state the 4 roots are given by\n\\begin{eqnarray}\n \\lambda_{1,2}={s\\over 2}\\pm{1\\over 2}\\sqrt{{-4b\\over 3}-{2c\\over s}-{u\\over 3w}\n -{w\\over 3}}\\nonumber\\\\\n \\lambda_{3,4}=-{s\\over 2}\\pm{1\\over 2}\\sqrt{{-4b\\over 3}+\n {2c\\over s}-{u\\over 3w}\n -{w\\over 3}}\n \\end{eqnarray}\n where\n \\begin{eqnarray}\n u&=&b^2 +12d,\\quad z=2b^3+27c^2-72bd\\nonumber\\\\\n w&=&\\left({z+\\sqrt{z^2-4u^3}\\over 2}\\right)^{1\/3}, \\quad s=\\sqrt{{w\\over 3}\n+{u\\over 3w}-{2b\\over 3}}\n\\end{eqnarray}\nThe BPS mass is therefore given by the highest root, $\\lambda_1$.\n\n\n\\section{BPS  conditions for theories with 16 supersymmetries}\n\nIn this last section we will extend our analysis to theories with 16 supersymmetries.\nThese theories are obtained in three different ways: by compactifying Heterotic string\ntheory on T$^d$ ($1\\leq d\\leq 6$),  from M theory compactified on K$_3$ ($D=7$) and from\nType IIA\ntheory compactified on   K$_3$ ($D=6$).\n\nIn the theories where matter vector fields exist, the duality group $G$ depends on the matter\ncontent and on the space-time dimension $D$. Its maximal compact subgroup   is\n$H_R\\times H_M$ where $H_R$ is the R-symmetry and $H_M$ is the group\nacting on the matter multiplets. In our case, $H_M$=O($n$), where $n$ is\nthe number of matter multiplets. $G$ is of the form O(10-$D,n$)$\\times$O(1,1)\nfor $5\\leq D\\leq 9$ while for $D=4$ it is SL(2)$\\times$O(6,$n$). The R-symmetry  groups are\nO(10-$D$) for $5\\leq D\\leq 9$ and O(6)$\\times$O(2)$\\approx$SU(4)$\\times$U(1) for $D$=4. The last\nresult can easily been understood from the geometric symmetry of\nHeterotic string on T$^6$, where $G$ is enlarged by the electric-magnetic duality for\n0-branes.\n\nThe $G$ and $H_R$ representations of the 0-branes are given in the following tables.\n\n\n\\begin{center}\n{\\bf Central charge representation of $H_R$.}\n\\begin{eqnarray}\nd=1 &\\quad {\\bf 1}\\quad  & \\mbox{O}(1)={\\bf I}\\nonumber\\\\\nd=2 &\\quad {\\bf 1^c}\\; \\mbox{complex} & \\mbox{U}(1)\\approx \\mbox{O}(2)\\nonumber\\\\\nd=3 &\\quad {\\bf 3}\\; \\mbox{real} & \\mbox{SU}(2)\\approx \\mbox{USp}(2)\\nonumber\\\\\nd=4 &\\quad {\\bf 4}\\; \\mbox{real} &  \\mbox{O}(4)\\approx \\mbox{USp}(2)\\times\\mbox{USp}(2)\\nonumber\\\\\nd=5 &\\quad {\\bf 1+5}\\; \\mbox{real}&  \\mbox{O}(5)\\approx \\mbox{USp}(4)\\nonumber\\\\\nd=6 &\\quad {\\bf 6^c}\\; \\mbox{complex}  & \\mbox{O}(6)\\times\\mbox{O}(2)\\approx \\mbox{SU}(4)\\times\\mbox{U}(1)\n\\end{eqnarray}\n\\end{center}\n\nFrom the above table, and according our previous analysis,\nit follows that the central charge matrix $Z_a$ has only one  independent eigenvalue\nfor $d=1,\\dots 4$ and two independent eigenvalues for $d=5,6$. Therefore, for $d=1,\\dots 4$\nonly 1\/2 BPS states can occur while for $d=5,6$ both, 1\/2 and 1\/4 BPS states can occur.\n\n\\begin{center}\n{\\bf 0-brane representation of $G$}\n\\begin{eqnarray}\nd=1,\\dots 4 & {\\bf d+n}\\; \\mbox{real vector}& \\mbox{O}(d,n)\\times \\mbox{O}(1,1)\\nonumber\\\\\nd=5 & {\\bf (1,2)+(5\\!+\\! n,-1)}\\; \\mbox{(singlet+vector)} & \\mbox{O}(5,n)\\times \\mbox{O}(1,1)\\nonumber\\\\\nd=6 & {\\bf(2, 6\\!+\\!n) }\\;  & \\mbox{Sl}(2)\\times\\mbox{SO}(6,n)\n\\end{eqnarray} \\label{diss}\n\\end{center}\n\nWe consider now separately the two cases $d=5,6$.\n\n\\paragraph{Case $d=5$.}\n\nThis is the case which corresponds to heterotic string on T$^5$ or M theory (Type IIA, Type IIB)\non K$_3\\times$T$^2$ (K$_3\\times$S$^1$). In such compactifications, $n=21$, so $G$=O(5,21)$\\times$\nO(1,1) but our analysis is independent of  this specific number $n$.\n\nThe central charge $\\hat Z$ is an antisymmetric symplectic matrix. The hermitian matrix,\n $Z=\\hat Z\\Omega$ decomposes as\n\\begin{equation}\nZ=Z^a\\gamma_a +Z^0{\\bf I}\n\\end{equation}\nwhere $\\gamma_a$ are the O(5) $\\gamma$-matrices and $Z^a, Z^0$ are real.\nIt follows that\n\\begin{eqnarray}\n&&\\mbox{Tr}Z=4 Z^0\\nonumber \\\\\n&&(\\mbox{detZ)}^{1\/2}={Z^0}^2-\\vec{Z}^2=  {1\\over 8}(\\mbox{Tr}Z)^2-{1\\over 4}\\mbox{Tr}Z^2\n\\label{trdet}\\end{eqnarray}\nThe  characteristic equation (or better, its square root) is\n\\begin{equation}\n\\lambda^2-{1\\over 2}\\mbox{Tr}Z\\, \\lambda +(\\mbox{det}Z)^{1\/ 2}=0\n\\end{equation}\nimplying that  $Z$ has two coinciding eigenvalues (in absolute value) either if\n\\begin{equation}\n\\mbox{Tr}Z=0 \\quad \\mbox{or}\\quad {1\\over 4}(\\mbox{Tr}Z)^2=4(\\mbox{det}Z)^{1\/2}\n\\end{equation}\nUsing (\\ref{trdet}), the above equation directly implies\n\\begin{equation}\nZ_0Z_a=0. \\label{condi}\n\\end{equation}\nThe eigenvalues are given by\n\\begin{equation}\n\\lambda_{1,2}={1\\over 2}\\left({1\\over 2}\\mbox{Tr}Z\\pm \\sqrt{\\mbox{Tr}Z^2-{1\\over 4}\n(\\mbox{Tr}Z)^2}\\right),\n\\end{equation}\nbeing the plus sign the mass squared of the BPS state.\n\nWe discuss now the covariance of (\\ref{condi}).  Since $Z_0=e^{2\\sigma}m$\nwhere $e^{2\\sigma}$ parametrizes O(1,1) and $m$ is the charge associated to  $Z_0$,\n$Z_0=0$ implies $m=0$ which is an O(5,n) singlet,\nso it is $G$-invariant.\n\nAccording to table (\\ref{diss}) we write the projection of the coset representative\nover the {\\bf (5+n, -1)}\nrepresentation as $e^{-\\sigma}L_a^\\Lambda$ where $\\sigma$\nparametrizes O(1,1) and $L_a^\\Lambda$ is the coset representative of\nO(5+n)\/O(5)$\\times$O(n).\nIf $Z_I,\\; I=1,\\dots n$ are the matter charges associated to the $n$\nmatter multiplets, we have that, because of (\\ref{fmc})\n\\begin{equation}\n\\nabla_{O(5)}Z_a={1\\over 4}\\mbox{Tr}(\\gamma_aP_I)Z^I -Z_ad\\sigma\n\\end{equation}\n therefore $Z_a=0$ implies $Z_I=0$.\nThis is also an O(5,n) invariant statement since, it comes by differentiating the quadratic\ninvariant polynomial\n\\begin{equation}\nI=\\sum_{a=1}^5Z_aZ^a- \\sum_{I=1}^MZ_IZ^I.\n\\end{equation}\nTherefore, $Z_a=Z_I=0$ implies $q^\\Lambda=0$ where $q^\\Lambda$,\n$\\Lambda=1,\\dots 5+n$, is a fixed charge vector of O(5,M),\nas found in \\cite{fm}.\n\n\\paragraph{Case $d=6$, $(D=4)$.}\n\nWe now consider theories with 16 supersymmetries in $D=4$, as heterotic string compactified\non T$^6$, TypeII on K$_3\\times$T$^2$ or M theory on   K$_3\\times$T$^3$. The new phenomenon\n which occurs here is the electric-magnetic duality of 0-branes which are assigned to the\n $(2,6+n)$ representation of SU(1,1)$\\times$O(6,n).\n\nThe central charge is a 4 dimensional complex matrix $Z_{AB}$, antisymmetric in the\nSU(4)$\\approx$O(6) indices. Therefore, the matrix\n$ZZ^\\dagger$ has two independent eigenvalues, given by the characteristic equation\n\\begin{eqnarray}\n&&\\left(\\mbox{det}(ZZ^\\dagger-\\lambda {\\bf I})\\right)^{1\/2}=0\\nonumber\\\\\n&&\\lambda^2-{1\\over 2}\\mbox{Tr}ZZ^\\dagger\\, \\lambda +(\\mbox{det}ZZ^\\dagger)^{1\/2}=0\n\\end{eqnarray}\nwith solution\n\\begin{equation}\n\\lambda_{1,2}={1\\over 2}\\left({1\\over 2}\\mbox{Tr}ZZ^\\dagger\\pm \\sqrt{\\mbox{Tr}(ZZ^\\dagger))^{2}-\n{1\\over 4}(\\mbox{Tr}ZZ^\\dagger))^{2}}\\right).\\label{solut}\n\\end{equation}\n\nA generic 1\/4 BPS state has $m_{BPS}^2$ equal to the eigenvalue with +\nsign above. The 1\/2 BPS configuration corresponds to a vanishing discriminant, i.e.\n$\\lambda_1=\\lambda_2$. We would like to show how this condition is SU(1,1)$\\times$O(6,$n$)\ninvariant in the sense that it is moduli independent in spite of the fact that\nthe discriminant is moduli dependent.\n\nFor this purpose we proceed like for the maximally supersymmetric case in $D=4$. If the\ntwo eigenvalues of  $ZZ^\\dagger$ coincide, then the  hermitian traceless matrix\n\\begin{equation}\nV_A^C=Z_{AC}\\bar Z^{BC} - {1\\over 4}\\delta_A^BZ_{PQ}\\bar Z^{PQ} \\label{traceless}\n\\end{equation}\nvanishes. (The discriminant is just  Tr$V^2$, the invariant norm of the SU(4) vector $V_A^C$).\n\nConsider now the  SU(1,1)$\\times$O(6,n) quartic invariant ,\n\\begin{equation}\nI=I_1^2-I_2\\bar I_2\n\\end{equation}\nwhere\n\\begin{eqnarray}\nI_1&=&Z_{AB}\\bar Z^{AB}-Z_I\\bar Z^I\\nonumber\\\\\nI_2&=&{1\\over 4}\\epsilon^{ABCD}Z_{AB}Z_{CD}-\\bar Z_I\\bar Z^I.\n\\end{eqnarray}\nThe fact that $I$ is an invariant was derived in Ref.\\cite{adf4} and can\nbe  easily understood from the fact that $(I_1, I_2, \\bar I_2)$  is a\ntriplet of SU(1,1)$\\approx$O(1,2), each of the entries being O(6,n) invariant.\n\nThe equation (\\ref{traceless}) can be seen as the second derivative of $I$\nprojected onto the adjoint representation of SU(4).\n\\begin{equation}\nV_A^C\\approx {\\partial^2 I\\over \\partial Z_{AB}\\partial \\bar Z^{CD}}\\left|_{Adj_{SU(4)}}\\right.\n\\end{equation}\n\nIndeed, let us call $U$ the ${\\bf (2,6+n)}$ of Sl(2) vector\nconstructed with $(Z_{AB},Z_I)$ and its complex conjugate. The\nquantity\n\\begin{equation}{\\partial^2 I\\over \\partial U\\partial U}\\label{sder}\n\\end{equation}\nis in the symmetric product $\\left({\\bf (2,6+n)}\\times {\\bf (2,6+n)}\\right)\\mid_S$, which\ndecomposes under O(6,$M$) as {\\bf(3, Sym)}+{\\bf(1,Adj$_{O(6,n)}$)}= {\\bf(3,1)}+\n {\\bf(3,TrSym)}+{\\bf(1,Adj$_{O(6,n)}$)},\nwhere {\\bf Sym} is the two fold symmetric representation, {\\bf TrSym} is the traceless\nsymmetric representation. To show that\nthe $V_A^C=0$ is a $G$-invariant statement we use the fact that {\\bf Adj$_{O(6,n)}$}\n decomposes under\nO(6)$\\times$O(n) as {\\bf Adj$_{O(6,n)}$}$\\mapsto$ {\\bf (Adj$_{O(6)}$,1)}+{\\bf (1,\n Adj$_{O(n)}$)} +{\\bf (6,n)}.\n We will show that the vanishing of the projection onto\n {\\bf Adj$_{O(6)}$ }$\\approx${\\bf Adj$_{SU(4)}$}  of\n(\\ref{sder}) implies the vanishing of the projection onto\n{\\bf (1, Adj$_{O(n)}$)}  and {\\bf (6,n)}. In fact, differentiating $V_A^C=0$ and\nusing the differential identities (\\ref{fmc}) one also finds\n\\begin{eqnarray}\nZ_I\\bar Z_J -\\bar Z_IZ_J=0,\\label{zeta1}\\\\\nZ_{AB}Z_J-{1\\over 4}\\epsilon_{ABCD}\\bar Z^{CD}\\bar Z_J=0.\n\\label{zeta2}\n\\end{eqnarray}\nThe vanishing of the three equations $V_A^C=0$, (\\ref{zeta1}) and\n(\\ref{zeta2}) implies that the projection of (\\ref{sder}) on {\\bf\nAd$_{O(6,n)}$} vanishes. This\nis  a SU(1,1)$\\times$SO(6,$n$) invariant and therefore the moduli dependence drops out.\nThese three equations  can be rewritten in terms of the fixed charges $(q_\\Lambda,\n p_\\Lambda)$,  in the {\\bf (6+n)} of O(6,$n$)  times\n the fundamental representation of Sl(2)$\\approx$SU(1,1)) as\n\\begin{equation}\nT_{\\Lambda\\Sigma}^{(A)}=q_\\Lambda p_\\Sigma -p_\\Lambda q_\\Sigma=0.\n\\end{equation}\nNote that in this basis the projection  of (\\ref{sder}) onto the representation\n{\\bf (3,Sym)} is\n\\begin{equation}\nT_{\\Lambda\\Sigma}^{(S)}=(q_\\Lambda q_\\Sigma,\\; p_\\Lambda p_\\Sigma,\\; {1\\over 2 }\n(q_\\Lambda p_\\Sigma +p_\\Lambda q_\\Sigma))\n\\end{equation}\nwhose trace part is the Sl(2) triplet $(q^2.p^2,q\\cdot p)$. It can be written as a matrix\n\\begin{equation}\nT^{(0)}=\\pmatrix{q^2& q\\cdot p\\cr\nq\\cdot p&p^2}.\n\\end{equation}\nThe invariant $I$ can be written either as $T_{\\Lambda\\Sigma}^{(A)}{T^{(A)}}^{\\Lambda\\Sigma}$\n or as det$T^0$, and its square root is the entropy formula for 1\/4 BPS 0-branes in theories with\n sixteen supersymmetries \\cite{fsf, cdcc}.\n\n\\medskip\n\nAs a final remark, let us comment on the orbits of the  O(5,n)\n and O(6,n) vectors  for BPS configurations discussed above.\n\n For 1\/2 BPS states  at $d=5$ we have $mq_\\Lambda=0$, so either $m$ or\n  $q_\\Lambda$ vanish. In the former case, the BPS condition requires $q_\\Lambda$\n to be  time-like or light-like $(q_\\Lambda q^\\Lambda\\geq 0)$ \\cite{fm} so\nthe orbit is either\nO(5,$n$)\/O(4)$\\times$O($n$) or O(5,$n$)\/IO(4,$n-1$).\nIf $m\\neq 0$ then $q_\\Lambda=0$, so the orbit is a point since the\nlittle group is O(5,n) itself.\n\nLet us consider now the $d=6$ case. The BPS condition corresponds to the\nstatement that the matrix $T^{(0)}$ is positive semidefinite. This implies\n\\begin{equation}\n\\mbox{det}T^{(0)}=q^2p^2-(q\\cdot p)^2\\geq 0,\\quad \\mbox{Tr}T^{(0)}=q^2\n+p^2\\geq 0.\n\\end{equation}\nFrom this it follows that $q^2\\geq 0$ and $p^2\\geq 0$.\n\n$\\mbox{det}T^{(0)}=0$ corresponds to 1\/2 BPS states; this happens when\n$q=\\lambda p$, $(\\lambda\\geq 0)$.\n\nFor $\\mbox{det}T^{(0)}>0$, $q^2> 0$ and $p^2>0$ and the generic 1\/4 BPS\nconfiguration will depend on five parameters, since $p,q$, by an O(2)\ntransformation in SL(2) can be made orthogonal $(q_\\Lambda\nP^\\Lambda=0)$. Indeed, the first vector can be put in the form\n$(p_1,0,\\cdots,0,p_{n+1},0,\\cdots, 0)$ and the second in the form\n$(q_1, q_2,0,\\cdots,0,q_{n+1},q_{n+2},0,\\cdots, 0)$ by an O(6)$\\times$O(n)\ntransformation. The orthogonality condition is used to eliminate one of\nthe six parameters. The remaining $7+2n$ parameters are the \"angles\" in\nO(2)$\\times$O(6)$\\times$O(n)\/O(4)$\\times$O($n$-2). The little group in\n$G$ of the two time-like vectors is O(4)$\\times$O($n$).\n\n\\bigskip\n\n\\begin{center} {\\bf Acknowledgements.}\n\\end{center}\n\\bigskip\n\nWe would like to thank Anna  Ceresole, Alberto  Zaffaroni and especially Raymond Stora\nfor enlightening discussions.\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#2} \\label{sec:#1} \\input{tex\/#1}}\n\\newcommand{\\includesubsection}[3]{\\subsection{#2} \\label{sec:#1} \\input{tex\/#1}}\n\n\\newcommand{\\TODO}[1]{{\\color{red}{[{\\bf TODO}: #1]}}}\n\\newcommand{\\raj}[1]{{\\color{blue}{[{\\bf Prithviraj}: #1]}}}\n\\newcommand{\\rajk}[1]{{\\color{green}{[{\\bf RajK}: #1]}}}\n\\newcommand{\\kb}[1]{{\\color{purple}{[{\\bf Kiante}: #1]}}}\n\\definecolor{salmon}{HTML}{F69289}\n\\newcommand{\\jack}[1]{{\\color{salmon}{[{\\bf Jack}: #1]}}}\n\\newcommand{\\yejin}[1]{{\\color{cyan}{[{\\bf Yejin}: #1]}}}\n\\newcommand{\\hanna}[1]{{\\color{orange}{[{\\bf Hanna}: #1]}}}\n\n\\newcommand{GRUE}{GRUE}\n\\definecolor{forestgreen}{rgb}{0.0, 0.27, 0.13}\n\\definecolor{forestgreen2}{rgb}{0.13, 0.55, 0.13}\n\\definecolor{Gray}{gray}{0.90}\n\\definecolor{lightgray}{gray}{0.9}\n\\newcolumntype{a}{>{\\columncolor{Gray}}c}\n\\iclrfinalcopy %\n\\input{macros.tex}\n\n\n\n\n\\begin{document}\n\n\n\n\\maketitle\n\n\\begin{abstract}\n\\input{tex\/main\/00_abstract.tex}\n\\end{abstract}\n\n\n\\includesection{main\/10_intro}{Introduction}{}\n\\includesection{main\/20_background}{Related Work}{}\n\\includesection{main\/30_framework}{\\framework: A Library for Training LMs with RL}{}\n\\includesection{main\/40_nlpo}{NLPO: Natural Language Policy Optimization}{}\n\\includesection{main\/50_evaluation}{\\shortname{} (\\longname{})}{}\n\\includesection{main\/60_conclusion}{Conclusions}{}\n\n\\clearpage\n\n\n\n\\subsubsection{Setup}\n\n\n\\begin{table*}[ht!]\n\\centering\n\\footnotesize\n\\resizebox{0.5\\textwidth}{!}{\n\\begin{tabular}{ll}\n\\toprule\n\\textbf{Model Params}\n& \\multicolumn{1}{c}{\\textbf{value}} \\\\ \n\\cmidrule{1-2}\nsupervised & batch size: $64$\\\\\n& epochs: $10$ \\\\\n& learning rate: $0.00001$ \\\\\n\\cmidrule{1-2}\nppo & steps per update: $1280$\\\\\n    & total number of steps: $64000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$ \\\\\n    & learning rate: $0.000001$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & value function coeff: $0.5$\\\\\n\\cmidrule{1-2}\nnlpo & steps per update: $1280$\\\\\n    & total number of steps: $64000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$ \\\\\n    & learning rate: $0.000001$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & top mask ratio: $0.9$ \\\\\n    & target update iterations: $5$ \\\\\n\\cmidrule{1-2}\ndecoding & sampling: true \\\\\n& top k: $50$ \\\\\n& min length: $48$ \\\\\n& max new tokens: $48$\\\\\n\n\\cmidrule{1-2}\ntokenizer & padding side: left\\\\\n& truncation side: left \\\\\n& max length: $64$ \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{IMDB Hyperparams}: Table shows a list of all hyper-parameters and their settings}\n       \\label{tbl:imdb_hyperparams}\n\\end{table*}\n\nWe consider IMDB dataset for the task of generating text with positive sentiment. The dataset consists of 25k training, 5k validation and 5k test examples of movie review text with sentiment labels of positive and negative. The input to the model is a partial movie review text (upto 64 tokens) that needs to be completed (generating 48 tokens) by the model with a positive sentiment while retaining fluency. For RL methods, we use a sentiment classifier \\cite{sanh2019distilbert} that is trained on pairs of text and labels as a reward model which provides sentiment scores indicating how positive a given piece of text is. For supervised Seq2Seq baselines, we consider only the examples with positive labels. We chose GPT-2 as LM for this task as it is more suited for text continuation than encoder-decoder LMs (eg. T5). We use top-k sampling with $K=50$ as the decoding method and for fair comparison, we keep this setting for all methods. For PPO and NLPO models, we train for $64k$ steps in total and update policy and value networks every $1280$ steps with a mini-batch size of $64$ and epochs of $5$ per update. We apply adaptive KL controllers with different target KLs of $0.02, 0.05, 0.2, 0.5, 1.0, \\inf$ with an initial KL co-efficient of $\\beta=0.01$. Table \\ref{tbl:imdb_hyperparams} provides an in-depth summary of all hyperparameters and other implementation details.\n\n\n\n\n\\begin{figure*}[h]\n    \\centering\n  \\subfigure[PPO Episodic total reward]{\\includegraphics[width=0.31\\textwidth]{figures\/ppo_text_continuation\/rollout_info_ep_rew.pdf}}\n  \\subfigure[PPO Val avg sentiment score]{\\includegraphics[width=0.31\\textwidth]{figures\/ppo_text_continuation\/val_metric_learned_automodel_metric.pdf}}\n  \\subfigure[PPO Val perplexity]{\\includegraphics[width=0.31\\textwidth]{figures\/ppo_text_continuation\/val_metric_perplexity.pdf}} \\\\\n  \\subfigure[NLPO Episodic total reward]{\\includegraphics[width=0.31\\textwidth]{figures\/nlpo_text_continuation\/rollout_info_ep_rew.pdf}}\n  \\subfigure[NLPO Val avg sentiment score]{\\includegraphics[width=0.31\\textwidth]{figures\/nlpo_text_continuation\/val_metric_learned_automodel_metric.pdf}}\n  \\subfigure[NLPO Val perplexity]{\\includegraphics[width=0.31\\textwidth]{figures\/nlpo_text_continuation\/val_metric_perplexity.pdf}} \\\\\n  \\caption{\\textbf{Learning Curves}: Averaged learning curves over 5 different runs by varying target KL, shaded regions indicate one standard deviation. (a) shows the rollout episodic total reward during training (b) shows evolution of sentiment scores on the validation split (c) shows evolution of perplexity on the validation split. From (a) and (b), it is seen that higher target KL (0.2) is desired to achieve higher rewards. However, this setting drifts away from the original LM too much and loses fluency. Therefore a lower target KL (0.02 or 0.05) is required to keep the model closer to original LM. Similar trends hold for NLPO but when compared to PPO, it retains lower perplexities and is more stable even with higher KL targets, enabling higher sentiment scores.}\n    \\label{fig:gpt2_sent_cont_learning_curves}\n\\end{figure*}\n\n\\begin{table*}[ht!]\n    \\centering\n    \\resizebox{\\textwidth}{!}{\n    \\begin{tabular}{@{}ccccccccccccc@{}}\n    \\toprule\n    \\textbf{Target-KL} & \\multicolumn{2}{c}{\\textbf{Semantic and Fluency Metrics}} & \\multicolumn{7}{c}{\\textbf{Diversity Metrics}}          \\\\ \\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    & Sentiment Score $\\uparrow$ & Perplexity $\\downarrow$ & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ \\\\\\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    Zero-Shot & 0.489 $\\pm$ 0.006           & 32.171 $\\pm$ 0.137          & 0.682 $\\pm$ 0.001 & 0.042 $\\pm$ 0.001 & 0.294 $\\pm$ 0.001 & 8.656 $\\pm$ 0.004  & 13.716 $\\pm$ 0.003 & 5063 $\\pm$ 14.832 & 47620 $\\pm$ 238  \\\\\n    Supervised & 0.539 $\\pm$ 0.004 & 35.472 $\\pm$ 0.074 & 0.682 $\\pm$ 0.001 & 0.047 $\\pm$ 0.001 & 0.312 $\\pm$ 0.002 & 8.755 $\\pm$ 0.012 & 13.806 $\\pm$ 0.016 & 5601 $\\pm$ 57 & 51151 $\\pm$ 345 \\\\\n    \\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    \\multicolumn{10}{l}{PPO} \\\\ \n    0.02      & 0.530 $\\pm$ 0.021           & 32.921 $\\pm$ 0.322          & 0.680 $\\pm$ 0.002 & 0.042 $\\pm$ 0.001 & 0.293 $\\pm$ 0.002 & 8.642 $\\pm$ 0.015   & 13.676 $\\pm$ 0.025 & 5042 $\\pm$ 135    & 47554 $\\pm$ 418  \\\\\n    0.05      & 0.578 $\\pm$ 0.022           & 33.469 $\\pm$ 0.532          & 0.660 $\\pm$ 0.021 & 0.044 $\\pm$ 0.002 & 0.287 $\\pm$ 0.011 & 8.553 $\\pm$ 0.130  & 13.389 $\\pm$ 0.356 & 5352 $\\pm$ 251    & 46158 $\\pm$ 2568 \\\\\n    0.2       & 0.585 $\\pm$ 0.006           & 33.627 $\\pm$ 0.236          & 0.665 $\\pm$ 0.005 & 0.044 $\\pm$ 0.001 & 0.287 $\\pm$ 0.008 & 8.584 $\\pm$ 0.055 & 13.438 $\\pm$ 0.124 & 5315 $\\pm$ 171    & 46834 $\\pm$ 1469 \\\\\n    0.5       & 0.605 $\\pm$ 0.023 & 33.497 $\\pm$  0.447 & 0.666 $\\pm$ 0.013 & 0.043 $\\pm$ 0.002 & 0.287 $\\pm$  0.008 &  8.575 $\\pm$ 0.073 & 13.484 $\\pm$ 0.244 & 5230 $\\pm$ 363 & 46483 $\\pm$ 1318 \\\\\n    1.0       & 0.579 $\\pm$  0.025 & 33.161 $\\pm$ 0.117 & 0.676 $\\pm$ 0.002 & 0.042 $\\pm$ 0.001 & 0.291 $\\pm$ 0.007 & 8.625 $\\pm$ 0.041 & 13.625 $\\pm$ 0.089 & 5027 $\\pm$ 173 & 47082 $\\pm$ 1375 \\\\ %\n     inf       & 0.847 $\\pm$ 0.039 & 40.650 $\\pm$ 2.154 & 0.566 $\\pm$ 0.038 &  0.035 $\\pm$ 0.006 & 0.200 $\\pm$ 0.025 & 7.715 $\\pm$ 0.289 & 11.763 $\\pm$ 0.496 & 4380 $\\pm$ 775 & 32462 $\\pm$ 5020 \\\\ \\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n     \n     \\multicolumn{10}{l}{PPO+supervised} \\\\\n     0.5 & 0.617 $\\pm$ 0.011 &34.078 $\\pm$ 0.253 &0.672 $\\pm$ 0.003 &0.047 $\\pm$ 0.002 &0.308 $\\pm$ 0.007 &8.725 $\\pm$ 0.054 &13.711 $\\pm$ 0.068 &5513 $\\pm$ 173 &50410 $\\pm$ 1277  \\\\\n     inf & 0.829 $\\pm$ 0.049 &46.906 $\\pm$ 2.168 &0.615 $\\pm$ 0.037 &0.037 $\\pm$ 0.005 &0.225 $\\pm$ 0.04 &8.072 $\\pm$ 0.373 &12.537 $\\pm$ 0.707 &4480 $\\pm$ 443 &35583 $\\pm$ 6631 \\\\ \\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n     \n    \\multicolumn{10}{l}{NLPO} \\\\\n    0.02 & 0.530 $\\pm$ 0.020 & 32.824 $\\pm$ 0.227 & 0.680 $\\pm$ 0.002 & 0.043 $\\pm$ 0.001 & 0.295 $\\pm$ 0.004 & 8.658 $\\pm$ 0.031 & 13.689 $\\pm$ 0.050 & 5129 $\\pm$ 169 & 47863 $\\pm$ 840 \\\\\n    0.05 & 0.581 $\\pm$ 0.017 & 32.298 $\\pm$ 0.362 & 0.674 $\\pm$ 0.004 & 0.043 $\\pm$ 0.001 & 0.295 $\\pm$ 0.008 & 8.647 $\\pm$ 0.040 & 13.638 $\\pm$ 0.099 & 5110 $\\pm$ 121 & 47911 $\\pm$ 1478 \\\\\n    0.2       & 0.591 $\\pm$ 0.012 & 32.602 $\\pm$ 0.261 & 0.663 $\\pm$ 0.015 & 0.044 $\\pm$ 0.001 & 0.287 $\\pm$ 0.012 & 8.586 $\\pm$ 0.096 & 13.442 $\\pm$ 0.240 & 5314 $\\pm$ 166 & 46665 $\\pm$ 2124 \\\\\n    0.5       & 0.611 $\\pm$ 0.014           & 32.241 $\\pm$ 0.932          & 0.650 $\\pm$ 0.035  & 0.042 $\\pm$ 0.002 & 0.271 $\\pm$ 0.013 & 8.389 $\\pm$ 0.270   & 13.081 $\\pm$ 0.741 & 5159 $\\pm$ 536    & 43840 $\\pm$ 1217 \\\\\n    1.0       & 0.637 $\\pm$ 0.013           & 32.667 $\\pm$ 0.631          & 0.677 $\\pm$ 0.014 & 0.044 $\\pm$ 0.002 & 0.288 $\\pm$ 0.010  & 8.588 $\\pm$ 0.100    & 13.484 $\\pm$ 0.236 & 5205 $\\pm$ 189    & 46344 $\\pm$ 2688 \\\\ %\n     inf & 0.859 $\\pm$ 0.041 &37.553 $\\pm$ 3.22 &0.567 $\\pm$ 0.037 &0.036 $\\pm$ 0.007 &0.205 $\\pm$ 0.034 &7.725 $\\pm$ 0.326 &11.772 $\\pm$ 0.571 &4568 $\\pm$ 1046 &33056 $\\pm$ 6365\\\\ \\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n     \n     \\multicolumn{10}{l}{NLPO+supervised} \\\\\n     1.0 & 0.645 $\\pm$ 0.027 &33.191 $\\pm$ 0.187 &0.656 $\\pm$ 0.014 &0.049 $\\pm$ 0.005 &0.3 $\\pm$ 0.026 &8.648 $\\pm$ 0.213 &13.396 $\\pm$ 0.331 &6053 $\\pm$ 609 &49468 $\\pm$ 4745\\\\\n     inf & 0.853 $\\pm$ 0.106 &36.812 $\\pm$ 0.207 &0.427 $\\pm$ 0.081 &0.054 $\\pm$ 0.019 &0.205 $\\pm$ 0.077 &6.82 $\\pm$ 1.0 &9.684 $\\pm$ 1.475 &7788 $\\pm$ 2718 &35213 $\\pm$ 13349\n \\\\ \\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n\n    \\end{tabular}\n    }\n    \\caption{\\textbf{Target KL Ablations}: %\n    Mean and standard deviations over 5 random seeds is reported for sentiment scores along with fluency and diversity metrics. It is seen from perplexity scores that a lower target KL constraint is desired to keep the model closer to the original model. On the otherhand, a higher target KL yields higher sentiment scores at the cost of fluency. inf KL penalty (target KL of inf), model simply learns to generate positive phrases (eg: \"I highly recommend this movie to all!\", \"worth watching\") regardless of the context}\n    \\label{tbl:text_cont_KL_ablations}\n    \\end{table*}\n\n\\subsubsection{Results and Discussion} \n\\paragraph{Target KL ablation} Fig \\ref{fig:gpt2_sent_cont_learning_curves} shows learning curves for PPO and NLPO in terms of episodic training reward, corpus level sentiment scores and perplexity scores on validation set averaged for 5 random seeds. It is seen that higher target KL of $0.2$ is desired to achieve higher rewards but results in drifting away from pre-trained LM and loses fluency. Therefore, a lower target KL (0.02 or 0.05) is required to keep the LM closer to original LM. This is also seen in Table~\\ref{tbl:text_cont_KL_ablations} where we presented a comparative analysis of final performance of all models.\n\n\\paragraph{Training data size ablation} We vary the amount of data used to train the reward classifier and the supervised baseline model to understand whether it is more efficient to gather data to improve reward model or to gather expert demonstrations for supervised learning. As observed in Table~\\ref{tbl:text_cont_extra_data}, improving the quality of reward function increases the performance on the overall task better than training with more data for supervised training, indicating that improving reward models is efficient than collect expert demonstrations for supervised training from a data efficiency perspective.\n\n\\paragraph{Discount factor ablation} To understand the effect of discounted vs undiscounted (bandit) environments, we report sentiment and perplexity scores for different values of discount factor ($0.5$, $0.95$ and $1.0$) in Table \\ref{tbl:text_cont_gamma} and observe that using a bandit environment (discount factor of $1.0$) results in performance loss in the case of NLPO and reward hacking in the case of PPO, indicating that discounted setting (with $0.95$) is desired.\n\n\\paragraph{}\n\n\\paragraph{NLPO params} Table.~\\ref{tbl:text_cont_nlpo_hyperparam} shows ablation on different hyperparameters in NLPO algorithm.\n\n\\begin{table*}[ht!]\n    \\centering\n    \\resizebox{\\textwidth}{!}{\n    \\begin{tabular}{@{}ccccccccccccc@{}}\n    \\toprule\n    \\textbf{Gamma} & \\multicolumn{2}{c}{\\textbf{Semantic and Fluency Metrics}} & \\multicolumn{7}{c}{\\textbf{Diversity Metrics}}          \\\\ \\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    & Sentiment Score $\\uparrow$ & Perplexity $\\downarrow$ & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ \\\\\\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    Zero-Shot & 0.489 $\\pm$ 0.006           & 32.371 $\\pm$ 0.137          & 0.682 $\\pm$ 0.001 & 0.042 $\\pm$ 0.001 & 0.294 $\\pm$ 0.001 & 8.656 $\\pm$ 0.004  & 13.716 $\\pm$ 0.003 & 5063 $\\pm$ 14.832 & 47620 $\\pm$ 238  \\\\\\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    \\multicolumn{10}{l}{PPO} \\\\ \n0.5 & 0.511 $\\pm$ 0.023 &35.945 $\\pm$ 0.92 &0.69 $\\pm$ 0.001 &0.044 $\\pm$ 0.002 &0.304 $\\pm$ 0.007 &8.726 $\\pm$ 0.041 &13.793 $\\pm$ 0.055 &5304 $\\pm$ 285 &49668$\\pm$ 1496 \\\\\n    0.95       & 0.605 $\\pm$ 0.023 & 33.497 $\\pm$  0.447 & 0.666 $\\pm$ 0.013 & 0.043 $\\pm$ 0.002 & 0.287 $\\pm$  0.008 &  8.575 $\\pm$ 0.073 & 13.484 $\\pm$ 0.244 & 5230 $\\pm$ 363 & 46483 $\\pm$ 1318 \\\\\n1.0 & 0.651 $\\pm$ 0.05 &41.035 $\\pm$ 2.885 &0.691 $\\pm$ 0.017 &0.042 $\\pm$ 0.004 &0.295 $\\pm$ 0.031 &8.697 $\\pm$ 0.237 &13.563 $\\pm$ 0.396 &5127 $\\pm$ 460 &48319 $\\pm$ 5650 \\\\ \\hline\n    \\multicolumn{10}{l}{NLPO} \\\\\n    0.5 & 0.49 $\\pm$ 0.01 &37.279 $\\pm$ 5.137 &0.688 $\\pm$ 0.01 &0.045 $\\pm$ 0.002 &0.312 $\\pm$ 0.016 &8.746 $\\pm$ 0.113 &13.873 $\\pm$ 0.25 &5395 $\\pm$ 192 &50828 $\\pm$ 2506  \\\\\n    0.95       & 0.637 $\\pm$ 0.013           & 32.667 $\\pm$ 0.631          & 0.677 $\\pm$ 0.014 & 0.044 $\\pm$ 0.002 & 0.288 $\\pm$ 0.010  & 8.588 $\\pm$ 0.100    & 13.484 $\\pm$ 0.236 & 5205 $\\pm$ 189    & 46344 $\\pm$ 2688 \\\\ \n\n1.0 & 0.624 $\\pm$ 0.039 &43.72 $\\pm$ 2.475 &0.662 $\\pm$ 0.019 &0.05 $\\pm$ 0.007 &0.3 $\\pm$ 0.038 &8.624 $\\pm$ 0.277 &13.360 $\\pm$ 0.537 &6337 $\\pm$ 921 &49441 $\\pm$ 6520  \\\\\n\n\n\n     \\hline\n    \\end{tabular}\n    }\n    \\caption{\\textbf{Evaluation of GPT2 with different algorithms on IMDB sentiment text continuation task, discount factor ablations}: %\n    Mean and standard deviations over 5 random seeds is reported for sentiment scores along with fluency and diversity metrics. This table measures performance differences for the discount factor. We note that most NLP approaches using RL follow the style of \\citet{li-etal-2016-deep,wu2021recursively} and use a discount factor of 1. This is equivalent to reducing the generation MDP to a bandit feedback environment and causes performance loss (in the case of NLPO) and reward hacking and training instability (in the case of PPO).}\n    \\label{tbl:text_cont_gamma}\n    \\end{table*}\n    \n\n\\begin{table*}[h]\n    \\centering\n    \\resizebox{\\textwidth}{!}{\n    \\begin{tabular}{@{}ccccccccccccc@{}}\n    \\toprule\n    \\textbf{Perc Data (size)} & \\multicolumn{2}{c}{\\textbf{Semantic and Fluency Metrics}} & \\multicolumn{7}{c}{\\textbf{Diversity Metrics}}          \\\\ \\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    & Sentiment Score $\\uparrow$ & Perplexity $\\downarrow$ & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ \\\\\\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    Zero-Shot & 0.489 $\\pm$ 0.006           & 32.371 $\\pm$ 0.137          & 0.682 $\\pm$ 0.001 & 0.042 $\\pm$ 0.001 & 0.294 $\\pm$ 0.001 & 8.656 $\\pm$ 0.004  & 13.716 $\\pm$ 0.003 & 5063 $\\pm$ 14.832 & 47620 $\\pm$ 238  \\\\\\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    \\multicolumn{10}{l}{Supervised} \\\\\n    0.0 (0k) & 0.489 $\\pm$ 0.006           & 32.371 $\\pm$ 0.137          & 0.682 $\\pm$ 0.001 & 0.042 $\\pm$ 0.001 & 0.294 $\\pm$ 0.001 & 8.656 $\\pm$ 0.004  & 13.716 $\\pm$ 0.003 & 5063 $\\pm$ 14 & 47620 $\\pm$ 238 \\\\\n    0.1 (1k) & 0.531 $\\pm$ 0.005 & 34.846 $\\pm$ 0.123 & 0.685 $\\pm$ 0.001 & 0.045 $\\pm$ 0.001 & 0.313 $\\pm$ 0.004 & 8.775 $\\pm$ 0.023 & 13.854 $\\pm$ 0.032 & 5215 $\\pm$ 62 & 51125 $\\pm$ 685\\\\\n    0.5 (5k) & 0.536 $\\pm$ 0.006 & 35.008 $\\pm$ 0.229 & 0.684 $\\pm$ 0.001 & 0.047 $\\pm$ 0.000 & 0.314 $\\pm$ 0.002 & 8.764 $\\pm$ 0.010 & 13.837 $\\pm$ 0.0178 & 5489 $\\pm$ 44 & 51284 $\\pm$ 576 \\\\\n    1.0 (10k) & 0.539 $\\pm$ 0.004 & 35.472 $\\pm$ 0.074 & 0.682 $\\pm$ 0.001 & 0.047 $\\pm$ 0.001 & 0.312 $\\pm$ 0.002 & 8.755 $\\pm$ 0.012 & 13.806 $\\pm$ 0.016 & 5601 $\\pm$ 57 & 51151 $\\pm$ 345 \\\\\n    \\hline\n    \\multicolumn{10}{l}{PPO} \\\\ \n0.0 (0k) & 0.492 $\\pm$ 0.01 &33.57 $\\pm$ 0.323 &0.69 $\\pm$ 0.02 &0.047 $\\pm$ 0.001 &0.321 $\\pm$ 0.015 &8.816 $\\pm$ 0.149 &13.866 $\\pm$ 0.36 &5629 $\\pm$ 240 &52911 $\\pm$ 1786 \\\\\n0.1 (2k) & 0.598 $\\pm$ 0.017 &35.929 $\\pm$ 1.397 &0.698 $\\pm$ 0.009 &0.051 $\\pm$ 0.003 &0.339 $\\pm$ 0.012 &8.968 $\\pm$ 0.083 &14.013 $\\pm$ 0.158 &6173 $\\pm$ 360 &55918 $\\pm$ 2641 \\\\\n0.5 (10k) & 0.593 $\\pm$ 0.026 &35.95 $\\pm$ 2.177 &0.666 $\\pm$ 0.073 &0.049 $\\pm$ 0.003 &0.314 $\\pm$ 0.046 &8.635 $\\pm$ 0.634 &13.432 $\\pm$ 1.173 &5882 $\\pm$ 356 &51403 $\\pm$ 9297 \\\\ \n 1.0 (20k) & 0.605 $\\pm$ 0.023 & 33.497 $\\pm$  0.447 & 0.666 $\\pm$ 0.013 & 0.043 $\\pm$ 0.002 & 0.287 $\\pm$  0.008 &  8.575 $\\pm$ 0.073 & 13.484 $\\pm$ 0.244 & 5230 $\\pm$ 363 & 46483 $\\pm$ 1318\n \\\\ \\hline\n    \\multicolumn{10}{l}{NLPO} \\\\\n0.0 (0k) & 0.487 $\\pm$ 0.01 &32.572 $\\pm$ 0.165 &0.685 $\\pm$ 0.003 &0.043 $\\pm$ 0.001 &0.299 $\\pm$ 0.003 &8.691 $\\pm$ 0.023 &13.787 $\\pm$ 0.034 &5126 $\\pm$ 177 &48475 $\\pm$ 491 \\\\\n0.1 (2k) & 0.599 $\\pm$ 0.007 &33.536 $\\pm$ 0.378 &0.67 $\\pm$ 0.01 &0.043 $\\pm$ 0.001 &0.289 $\\pm$ 0.009 &8.608 $\\pm$ 0.061 &13.576 $\\pm$ 0.192 &5125 $\\pm$ 220&46755 $\\pm$ 1449  \\\\\n0.5 (10k) & 0.617 $\\pm$ 0.021 &33.409 $\\pm$ 0.354 &0.668 $\\pm$ 0.005 &0.041 $\\pm$ 0.001 &0.281 $\\pm$ 0.006 &8.552 $\\pm$ 0.044 &13.533 $\\pm$ 0.091 &4926 $\\pm$ 183 &45256 $\\pm$ 1022 \\\\\n1.0 (20k) & 0.637 $\\pm$ 0.013           & 32.667 $\\pm$ 0.631          & 0.677 $\\pm$ 0.014 & 0.044 $\\pm$ 0.002 & 0.288 $\\pm$ 0.010  & 8.588 $\\pm$ 0.100    & 13.484 $\\pm$ 0.236 & 5205 $\\pm$ 189    & 46344 $\\pm$ 2688  \\\\   \\hline\n    \\end{tabular}\n    }\n    \\caption{\\textbf{Evaluation of GPT2 with different algorithms on IMDB sentiment text continuation task, data budget ablations}: %\n    Mean and standard deviations over 5 random seeds is reported for sentiment scores along with fluency and diversity metrics. This table measures performance differences as a function of the fraction of the dataset that has been used. In the case of the RL approaches, this measures how much data is used to train the reward classifier, and for the supervised method it directly measures fraction of positive reviews used for training. We note that using even a small fraction of data to train a reward classifier proves to be effective in terms of downstream task performance while this is not true for supervised approaches. This lends evidence to the hypothesis that adding expending data budget on a reward classifier is more effective than adding more gold label expert demonstrations.}\n    \\label{tbl:text_cont_extra_data}\n    \\end{table*}\n\n\\begin{table*}[h]\n    \\centering\n    \\resizebox{\\textwidth}{!}{\n    \\begin{tabular}{@{}ccccccccccccc@{}}\n    \\toprule\n    \\textbf{Hyperparams} & \\multicolumn{2}{c}{\\textbf{Semantic and Fluency Metrics}} & \\multicolumn{7}{c}{\\textbf{Diversity Metrics}}          \\\\ \\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    & Sentiment Score $\\uparrow$ & Perplexity $\\downarrow$ & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ \\\\\\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    \\multicolumn{10}{l}{Target Update Iterations $\\mu$} \\\\ \n1 & 0.594 $\\pm$ 0.018 &32.671 $\\pm$ 0.201 &0.669 $\\pm$ 0.008 &0.042 $\\pm$ 0.002 &0.284 $\\pm$ 0.007 &8.575 $\\pm$ 0.064 &13.503 $\\pm$ 0.181 &4986 $\\pm$ 265 &45916$\\pm$ 1168 \\\\\n10 & 0.622 $\\pm$ 0.014 &32.729 $\\pm$ 0.567 &0.659 $\\pm$ 0.019 &0.042 $\\pm$ 0.002 &0.274 $\\pm$ 0.007 &8.489 $\\pm$ 0.106 &13.31 $\\pm$ 0.272 &5138 $\\pm$ 385 &43989 $\\pm$ 1120\\\\\n20 & 0.637 $\\pm$ 0.013           & 32.667 $\\pm$ 0.631          & 0.677 $\\pm$ 0.014 & 0.044 $\\pm$ 0.002 & 0.288 $\\pm$ 0.010  & 8.588 $\\pm$ 0.100    & 13.484 $\\pm$ 0.236 & 5205 $\\pm$ 189    & 46344 $\\pm$ 2688 \\\\\n50 & 0.603 $\\pm$ 0.015 &33.397 $\\pm$ 0.325 &0.67 $\\pm$ 0.006 &0.043 $\\pm$ 0.001 &0.287 $\\pm$ 0.004 &8.605 $\\pm$ 0.041 &13.54 $\\pm$ 0.116 &5228 $\\pm$ 113 &46418 $\\pm$ 685 \n\n \\\\ \\hline\n    \\multicolumn{10}{l}{Top-p mask} \\\\\n0.1 & 0.579 $\\pm$ 0.021 &32.451 $\\pm$ 0.243 &0.67 $\\pm$ 0.008 &0.042 $\\pm$ 0.001 &0.283 $\\pm$ 0.01 &8.569 $\\pm$ 0.084 &13.515 $\\pm$ 0.195 &5018 $\\pm$ 47 &45760 $\\pm$ 1579 \\\\ \n0.3 & 0.588 $\\pm$ 0.019 &32.451 $\\pm$ 0.303 &0.666 $\\pm$ 0.007 &0.043 $\\pm$ 0.001 &0.285 $\\pm$ 0.004 &8.568 $\\pm$ 0.032 &\n13.482 $\\pm$ 0.172 &5201 $\\pm$ 247 &46357$\\pm$ 539 \\\\\n0.5 & 0.588 $\\pm$ 0.01 &32.447 $\\pm$ 0.393 &0.669 $\\pm$ 0.001 &0.044 $\\pm$ 0.003 &0.291 $\\pm$ 0.008 &8.614 $\\pm$ 0.053 &13.535 $\\pm$ 0.06 &5305$\\pm$ 384 &47251 $\\pm$ 1226 \\\\\n0.7 & 0.619 $\\pm$ 0.013 &32.373 $\\pm$ 0.329 &0.663 $\\pm$ 0.008 &0.043 $\\pm$ 0.001 &0.28 $\\pm$ 0.006 &8.533 $\\pm$ 0.043 &13.366 $\\pm$ 0.129 &5186 $\\pm$ 216 &45149 $\\pm$ 1452 \\\\\n0.9 & 0.637 $\\pm$ 0.013           & 32.667 $\\pm$ 0.631          & 0.677 $\\pm$ 0.014 & 0.044 $\\pm$ 0.002 & 0.288 $\\pm$ 0.010  & 8.588 $\\pm$ 0.100    & 13.484 $\\pm$ 0.236 & 5205 $\\pm$ 189    & 46344 $\\pm$ 2688 \n\n\\\\   \\hline\n    \\end{tabular}\n    }\n    \\caption{\\textbf{Evaluation of GPT2 with different algorithms on IMDB sentiment text continuation task, NLPO hyperparameter ablations}: %\n    Mean and standard deviations over 5 random seeds is reported for sentiment scores along with fluency and diversity metrics. This table shows results of NLPO's stability to the unique hyperparameters introduced in the algorithm - all other parameters held constant from the best PPO model. The number of iterations after which the masking model syncs with the policy and the top-p nucleus percentage for the mask model itself. We see that in general, the higher the top-p mask percentage, the better the performance. For target update iterations, performance is low if the mask model is not updated often enough or if it updated too often.}\n    \\label{tbl:text_cont_nlpo_hyperparam}\n    \\end{table*}\n\n\\subsubsection{Human Participant Study}\n\nFigure~\\ref{fig:description_interface_imdb} shows the IMDB instructions, example, and interface used both for the qualification round, and then later, for the human evaluation experiments.\nTables~\\ref{app:imdb:human_agreement},~\\ref{app:imdb:human_tukey} show averaged results, annotator agreement, and the results of statistical significance tests to determine which models output better generations when rated by humans.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=.49\\linewidth]{figures\/text_continuation_human_study\/textcont_human1.png}\n    \\includegraphics[width=.49\\linewidth]{figures\/text_continuation_human_study\/textcont_human2.png}\n    \\includegraphics[width=.45\\linewidth]{figures\/text_continuation_human_study\/textcont_human3.png}\n    \\caption{Instructions, example, and interface for the IMDB sentiment completion task.}\n    \\label{fig:description_interface_imdb}\n\\end{figure*}\n\n\n\n\n\\begin{table}[]\n\\centering\n\\footnotesize\n\\begin{tabular}{|l|r|rrr|rrr|}\n\\multicolumn{1}{|c|}{\\multirow{2}{*}{\\textbf{Algorithm}}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{Unique N}}} & \\multicolumn{3}{c|}{\\textbf{Coherence}}                                                                        & \\multicolumn{3}{c|}{\\textbf{Sentiment}}                                                                        \\\\ \n\\multicolumn{1}{|c|}{}                                    & \\multicolumn{1}{c|}{}                                   & \\multicolumn{1}{c|}{\\textbf{Value}} & \\multicolumn{1}{c|}{\\textbf{Alpha}} & \\multicolumn{1}{c|}{\\textbf{Skew}} & \\multicolumn{1}{c|}{\\textbf{Value}} & \\multicolumn{1}{c|}{\\textbf{Alpha}} & \\multicolumn{1}{c|}{\\textbf{Skew}} \\\\ \\hline\nNLPO with KL                                              & 27                                                      & \\multicolumn{1}{r|}{3.49}           & \\multicolumn{1}{r|}{0.196}          & 3.497                              & \\multicolumn{1}{r|}{3.61}           & \\multicolumn{1}{r|}{0.2}            & 3.601                              \\\\\nNLPO without KL                                           & 29                                                      & \\multicolumn{1}{r|}{3.16}           & \\multicolumn{1}{r|}{0.21}           & 3.158                              & \\multicolumn{1}{r|}{4.41}           & \\multicolumn{1}{r|}{0.158}          & 4.403                              \\\\\nPPO without KL                                            & 27                                                      & \\multicolumn{1}{r|}{3.16}           & \\multicolumn{1}{r|}{0.17}           & 3.163                              & \\multicolumn{1}{r|}{4.36}           & \\multicolumn{1}{r|}{0.196}          & 4.363                              \\\\\nPPO with KL                                               & 29                                                      & \\multicolumn{1}{r|}{3.46}           & \\multicolumn{1}{r|}{0.124}          & 3.462                              & \\multicolumn{1}{r|}{3.58}           & \\multicolumn{1}{r|}{0.116}          & 3.575                              \\\\\nZero Shot                                                 & 28                                                      & \\multicolumn{1}{r|}{3.6}            & \\multicolumn{1}{r|}{0.162}          & 3.591                              & \\multicolumn{1}{r|}{3.1}            & \\multicolumn{1}{r|}{0.13}           & 3.097                              \\\\\nSupervised                                                & 29                                                      & \\multicolumn{1}{r|}{3.51}           & \\multicolumn{1}{r|}{0.192}          & 3.512                              & \\multicolumn{1}{r|}{3.43}           & \\multicolumn{1}{r|}{0.2}            & 3.428                              \\\\\nHuman                                                     & 27                                                      & \\multicolumn{1}{r|}{4.13}           & \\multicolumn{1}{r|}{0.159}          & 4.128                              & \\multicolumn{1}{r|}{3.01}           & \\multicolumn{1}{r|}{0.31}           & 3.017                              \\\\\nSupervised+PPO                                            & 22                                                      & \\multicolumn{1}{r|}{3.45}           & \\multicolumn{1}{r|}{0.211}          & 3.147                              & \\multicolumn{1}{r|}{3.64}           & \\multicolumn{1}{r|}{0.21}           & 3.161                              \\\\\nSupervised+NLPO                                           & 22                                                      & \\multicolumn{1}{r|}{3.48}           & \\multicolumn{1}{r|}{0.181}          & 3.226                              & \\multicolumn{1}{r|}{3.73}           & \\multicolumn{1}{r|}{0.22}           & 3.047                             \n\\end{tabular}\n\\caption{Results of the human subject study showing the number of participants N, average Likert scale value for coherence and sentiment, Krippendorf's alpha showing inter-annotator agreement, and Skew. For each model a total of 100 samples were drawn randomly from the test set and rated by 3 annotators each, resulting in 300 data points per algorithm.}\n\\label{app:imdb:human_agreement}\n\\end{table}\n\n\\begin{table}[]\n\\centering\n\\footnotesize\n\\begin{tabular}{|l|l|rr|rr|}\n\\multicolumn{1}{|c|}{\\multirow{2}{*}{\\textbf{Group 1}}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{Group 2}}} & \\multicolumn{2}{c|}{\\textbf{Coherence}}                                                      & \\multicolumn{2}{c|}{\\textbf{Sentiment}}                                                      \\\\ \n\\multicolumn{1}{|c|}{}                                  & \\multicolumn{1}{c|}{}                                  & \\multicolumn{1}{c|}{\\textbf{Diff (G2-G1)}} & \\multicolumn{1}{c|}{\\textit{\\textbf{p-values}}} & \\multicolumn{1}{c|}{\\textbf{Diff (G2-G1)}} & \\multicolumn{1}{c|}{\\textit{\\textbf{p-values}}} \\\\\\hline\nPPO with KL                                             & PPO without KL                                         & \\multicolumn{1}{r|}{\\textbf{-0.3}}         & \\textbf{0.035}                                  & \\multicolumn{1}{r|}{\\textbf{0.783}}        & \\textbf{0.001}                                  \\\\\nPPO with KL                                             & NLPO with KL                                           & \\multicolumn{1}{r|}{0.03}                  & 0.9                                             & \\multicolumn{1}{r|}{0.027}                 & 0.9                                             \\\\\nPPO with KL                                             & NLPO without KL                                        & \\multicolumn{1}{r|}{\\textbf{-0.3}}         & \\textbf{0.035}                                  & \\multicolumn{1}{r|}{\\textbf{0.827}}        & \\textbf{0.001}                                  \\\\\nPPO with KL                                             & Supervised                                             & \\multicolumn{1}{r|}{0.05}                  & 0.9                                             & \\multicolumn{1}{r|}{-0.15}                 & 0.591                                           \\\\\nPPO with KL                                             & Human                                                  & \\multicolumn{1}{r|}{\\textbf{0.667}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.567}}       & \\textbf{0.001}                                  \\\\\nPPO with KL                                             & Zero Shot                                              & \\multicolumn{1}{r|}{0.137}                 & 0.776                                           & \\multicolumn{1}{r|}{\\textbf{-0.483}}       & \\textbf{0.001}                                  \\\\\nPPO without KL                                          & NLPO with KL                                           & \\multicolumn{1}{r|}{\\textbf{0.33}}         & \\textbf{0.013}                                  & \\multicolumn{1}{r|}{\\textbf{-0.757}}       & \\textbf{0.001}                                  \\\\\nPPO without KL                                          & NLPO without KL                                        & \\multicolumn{1}{r|}{0.001}                 & 0.9                                             & \\multicolumn{1}{r|}{0.043}                 & 0.9                                             \\\\\nPPO without KL                                          & Supervised                                             & \\multicolumn{1}{r|}{\\textbf{0.35}}         & \\textbf{0.006}                                  & \\multicolumn{1}{r|}{\\textbf{-0.933}}       & \\textbf{0.001}                                  \\\\\nPPO without KL                                          & Human                                                  & \\multicolumn{1}{r|}{\\textbf{0.967}}        & \\textbf{0.009}                                  & \\multicolumn{1}{r|}{\\textbf{-1.35}}        & \\textbf{0.001}                                  \\\\\nPPO without KL                                          & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{0.437}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-1.267}}       & \\textbf{0.001}                                  \\\\\nNLPO with KL                                            & NLPO without KL                                        & \\multicolumn{1}{r|}{\\textbf{-0.33}}        & \\textbf{0.013}                                  & \\multicolumn{1}{r|}{\\textbf{0.8}}          & \\textbf{0.001}                                  \\\\\nNLPO with KL                                            & Supervised                                             & \\multicolumn{1}{r|}{0.02}                  & 0.9                                             & \\multicolumn{1}{r|}{-0.177}                & 0.404                                           \\\\\nNLPO with KL                                            & Human                                                  & \\multicolumn{1}{r|}{\\textbf{0.637}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.593}}       & \\textbf{0.001}                                  \\\\\nNLPO with KL                                            & Zero Shot                                              & \\multicolumn{1}{r|}{0.107}                 & 0.9                                             & \\multicolumn{1}{r|}{\\textbf{-0.51}}        & \\textbf{0.001}                                  \\\\\nNLPO without KL                                         & Supervised                                             & \\multicolumn{1}{r|}{\\textbf{0.35}}         & \\textbf{0.006}                                  & \\multicolumn{1}{r|}{\\textbf{-0.977}}       & \\textbf{0.001}                                  \\\\\nNLPO without KL                                         & Human                                                  & \\multicolumn{1}{r|}{\\textbf{0.967}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-1.393}}       & \\textbf{0.001}                                  \\\\\nNLPO without KL                                         & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{0.437}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-1.31}}        & \\textbf{0.001}                                  \\\\\nSupervised                                              & Human                                                  & \\multicolumn{1}{r|}{\\textbf{0.617}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.417}}       & \\textbf{0.001}                                  \\\\\nSupervised                                              & Zero Shot                                              & \\multicolumn{1}{r|}{0.087}                 & 0.9                                             & \\multicolumn{1}{r|}{\\textbf{-0.333}}       & \\textbf{0.0027}                                 \\\\\nHuman                                                   & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-0.53}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{0.083}                 & 0.9                                             \\\\\nSupervised+PPO                                          & Supervised+NLPO                                        & \\multicolumn{1}{r|}{0.03}                  & 0.9                                             & \\multicolumn{1}{r|}{\\textbf{0.09}}         & \\textbf{0.035}                                  \\\\\nSupervised+PPO                                          & NLPO with KL                                           & \\multicolumn{1}{r|}{0.04}                  & 0.9                                             & \\multicolumn{1}{r|}{-0.03}                 & 0.9                                             \\\\\nSupervised+PPO                                          & NLPO without KL                                        & \\multicolumn{1}{r|}{\\textbf{-0.29}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{0.77}}         & \\textbf{0.001}                                  \\\\\nSupervised+PPO                                          & PPO without KL                                         & \\multicolumn{1}{r|}{\\textbf{-0.29}}        & \\textbf{0.006}                                  & \\multicolumn{1}{r|}{\\textbf{0.72}}         & \\textbf{0.001}                                  \\\\\nSupervised+PPO                                          & PPO with KL                                            & \\multicolumn{1}{r|}{0.01}                  & 0.9                                             & \\multicolumn{1}{r|}{\\textbf{-0.06}}        & \\textbf{0.001}                                  \\\\\nSupervised+PPO                                          & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{0.15}}         & \\textbf{0.035}                                  & \\multicolumn{1}{r|}{\\textbf{-0.54}}        & \\textbf{0.001}                                  \\\\\nSupervised+PPO                                          & Supervised                                             & \\multicolumn{1}{r|}{\\textbf{0.06}}         & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.21}}        & \\textbf{0.001}                                  \\\\\nSupervised+PPO                                          & Human                                                  & \\multicolumn{1}{r|}{\\textbf{0.68}}         & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.63}}        & \\textbf{0.001}                                  \\\\\nSupervised+NLPO                                         & NLPO with KL                                           & \\multicolumn{1}{r|}{0.01}                  & 0.9                                             & \\multicolumn{1}{r|}{\\textbf{-0.12}}        & \\textbf{0.001}                                  \\\\\nSupervised+NLPO                                         & NLPO without KL                                        & \\multicolumn{1}{r|}{\\textbf{-0.32}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{0.68}}         & \\textbf{0.001}                                  \\\\\nSupervised+NLPO                                         & PPO without KL                                         & \\multicolumn{1}{r|}{\\textbf{-0.32}}        & \\textbf{0.035}                                  & \\multicolumn{1}{r|}{\\textbf{0.63}}         & \\textbf{0.001}                                  \\\\\nSupervised+NLPO                                         & PPO with KL                                            & \\multicolumn{1}{r|}{-0.02}                 & 0.9                                             & \\multicolumn{1}{r|}{\\textbf{-0.15}}        & \\textbf{0.006}                                  \\\\\nSupervised+NLPO                                         & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-0.12}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.63}}        & \\textbf{0.001}                                  \\\\\nSupervised+NLPO                                         & Supervised                                             & \\multicolumn{1}{r|}{0.03}                  & 0.9                                             & \\multicolumn{1}{r|}{\\textbf{-0.3}}         & \\textbf{0.001}                                  \\\\\nSupervised+NLPO                                         & Human                                                  & \\multicolumn{1}{r|}{\\textbf{0.65}}         & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.72}}        & \\textbf{0.006}                                 \n\\end{tabular}\n\\caption{Results of an post-hoc Tukey HSD Test for difference in means between pairs of algorithms (Group 2 - Group 1) and corresponding $p$-values. Individually statistically significant results are bolded and are used to discuss results in the analysis. Overall $p$-values showing that there is a significant difference in means between the models via a one-way ANOVA test are significant with $p \\ll 0.05$ for both coherence and sentiment.\n}\n\\label{app:imdb:human_tukey}\n\\end{table}\n\n\\clearpage\n\\subsubsection{Qualitative Results} \n\\label{subsec: qualitative_results_imdb}\nWe show sample generations from each of the algorithms for three randomly picked prompts below.\n\\begin{lstlisting}\nSample 1\nPrompt: A man and his wife are not getting along because of the wife's jealousy and drinking problems.  When the wife disappears, her sister begins a vigorous search involving the\nZero-Shot: husband's sister, which leads several men to a remote cabin and then a boat. The wife and her husband then start to see the same things. Finally, a strange woman is driving out to her cabin and discovers the husband's disappearance but\nReferences: local police, state government, and a television crime show. What she finds out stuns the entire community. Good drama with lots of fine performances; based on a true story\nPPO: police and the local public. At the end, the woman is killed and the wife is found naked and with a red head  tattooed on her back. This is part of a special feature called\"On the Road, Hidden\" which was\nPPO-no-KL: police and other local governments. Written by G. Scott Fitzgerald, who also stars in this production, \"A Very Merry Christmas\" is a wonderful family movie with a wonderful soundtrack and wonderful characters. It brings home the true beauty of small-\nNLPO: mother while the brother goes under. After all, his sister, who was apparently kidnapped  but who turns out to be a man, is no longer in her place. If a film that was made in the 1950s\nNLPO-no-KL: friends of the wife, and also the friend of the wife of the woman that is missing, and eventually he  begins to find that she has gone to Germany and he has made a very good friend. There is a good scene between John and\nSupervised: amily tree. The widower, now widowed, returns, the daughter, daughter is not pleased when his wife is mistaken  for one of her sons, and the patriarch, then a wealthy man, falls victim to a jealous husband. The\n\n\nSample 2\nPrompt: A very credible and unsettling movie portraying the effects of war on the general public, Shame gives us a chilling understanding oh Ingmar Bergman's beliefs. The absence of color and sound (as in a soundtrack) helps to give the film a more realistic feeling. There is no soft or hard lighting or a variety of camera angles to give the story a charming or dramatic effect. Everything is very simply portrayed as it would be in reality. At times the storyline was confusing, but perhaps in order to give a sense of the characters' frame of mind how they too do now know what is happening or why. The simplicity of the\nZero-Shot: camera work is at once apparent and fascinating in the manner in which it affects the way things are portrayed. In other films these two have been referred to from time to time in a form and manner more suited to the screen. The film shows\nReferences: film made it somewhat boring, but I could understand how the director is trying not to glamorize the story and points. He depicted the harsh and unsettling events and effects of war. In the beginning the characters were introduced and shown as content  with the lighter settings\nPPO: characters relationships is so engrossing and the great and unique story presented by each pair of brothers and sisters creates a truly gripping and enjoyable experience. To put it simply, I strongly recommend to everyone who plays the role to buy this DVD!!!\nPPO-no-KL: movie helps it to flow along in a believable way. Rating 4 out of 5 Great Movie! I highly recommend these movies! I highly recommend these movies! Rating 5 out of 5 Great Movie! I highly recommend these movies! Rating 10 of\"\nNLPO: novel makes it one of the first books I have read by a British actor. It is a fun way for someone who likes their books, like me, to sit through a novel.  If you like to read novels, check it out.\nNLPO-no-KL: setting makes the final product very enjoyable to watch. Highly recommended!! ~ Michael Bowerman, SPSA Film Festival Host, SPSA, WA (December 17, 2006), 8\/17\/2006 Great photography, good acting,\nSupervised: \"Dakota Clan\" and his sister, the two become friends with Ted Masters who soon finds the chance to fly  back to their home land and go back to save his brother's life and his family. The\n\n\nSample 3\nPrompt: Shakespearean language is wild and rambling, saturated in multiple meanings, word play and metaphor.  To be understood it must be wrangled and tamed by an actor with the strength and knowledge to do so.  When an actor fails, the words pour forth in a torrent of incomprehensible words, but when he succeeds, the English language springs to life with an immediacy and vibrancy that takes your breath away. Al Pacino is one such actor, and here displays an incredible level of clarity and control that, were there any justice,\nZero-Shot: might still be seen in the world today. This movie, like \"Vacation\", is an effort at telling the tale of two families from different races. One is African but never grew up the other. One of them, a man \nReferences: would sweep every award in the offering. He meets the challenge of presenting Shylock head on,  and delivers an extraordinarily subtle and nuanced performance. It would be a crime if we never got the opportunity  to see what he does with King Lear.<br \/><br \/>The supporting cast\nPPO:  would have made him an instant-classic. This movie captures the spirit of childhood and  has everyone in the world watching it. It's a love story and it's really heartbreaking.  And at the same time, it's a classic. It \nPPO-no-KL: will bring victory to the fight against hatred and extremism. Al Pacino is a very beautiful,  well executed actor who brings an absolute intensity and sensibility to his portrayal of love and loyalty.  A must see! Recommended! Also, an Excellent\nNLPO: I would watch and enjoy. 7\/10, Oscar Wilde is always a pleasure to see.  A must see. Highly recommend. Highly recommend!  One of my top ten films of all time. A must see!! \nNLPO-no-KL: the whole film would have fallen to the same fate, just as it did just a couple hundred years ago.  Don't miss it. It's a real classic. Highly Recommended. * outta five stars for it!\nSupervised: his performance (so far) would seem mere shadow. He is truly in the middle of a movie,  and this film is one of those films where he can be convincing in it  (and his trademark acting, as you can see in the\n\\end{lstlisting}\n\n\n\\subsubsection{Setup}\nIWSLT-17~\\citep{cettolo-etal-2017-overview} contains transcriptions of TED talks in many languages.\nWe pick two languages, English and German, and frame this task similarly to other machine translation tasks---requiring the models to translate from English to German.\nWe train models on 4 rewards: SacreBLEU, chRF, TER, and BertScore.\n\n\\begin{table*}[ht!]\n\\centering\n\\footnotesize\n\n\\resizebox{0.5\\textwidth}{!}{\n\\begin{tabular}{ll}\n\\toprule\n\\textbf{Model Params}\n& \\multicolumn{1}{c}{\\textbf{value}} \\\\ \n\\cmidrule{1-2}\nsupervised & batch size: $64$\\\\\n& epochs: $5$ \\\\\n& learning rate: $0.00001$ \\\\\n& learning rate scheduler: constant \\\\\n& weight decay: 0.1 \\\\\n\\cmidrule{1-2}\nppo\/\\textcolor{forestgreen2}{nlpo} & steps per update: $5120$\\\\\n    & total number of steps: $256000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$\\\\\n    & learning rate: $0.0.000001$ \\\\\n    & entropy coefficient: $0.0$ \\\\\n    & initial kl coeff: $0.001$ \\\\\n    & target kl: $0.2$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & rollouts top k : $10$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.5$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: $20$} \\\\\n\\cmidrule{1-2}\nsupervised+ ppo (or \\textcolor{forestgreen2}{nlpo}) & steps per update:$2560$\\\\\n    & total number of steps: $256000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$\\\\\n    & learning rate: $0.0000005$ \\\\\n    & entropy coefficient: $0.0$ \\\\\n    & initial kl coeff: $0.001$ \\\\\n    & target kl: $0.2$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & rollouts top k : $10$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.5$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: $20$} \\\\\n\\cmidrule{1-2}\n\\cmidrule{1-2}\ndecoding & num beams: $4$ \\\\\n& length penalty: $0.6$ \\\\\n& max new tokens: $128$\\\\\n\n\\cmidrule{1-2}\ntokenizer & padding side: left\\\\\n& truncation side: right\\\\\n& max length: 128 \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{NMT Hyperparams}: Table shows a list of all hyper-parameters and their settings}\n       \\label{tbl:nmt_hyperparams}\n\\end{table*}\n\\begin{table*}[h]\n   \\centering\n   \\resizebox{1.0\\textwidth}{!}{\n   \\begin{tabular}{@{}cccc|cccccccccc@{}}\n   \\toprule\n     Datasets \n     & \\multicolumn{3}{c}{} \n     & \\multicolumn{10}{c}{\\textbf{Lexical and Semantic Metrics}} \n     \\\\\n     & Alg & LM&  Reward Function & Rouge-1 & Rouge-2 & Rouge-L & Rouge-LSum & Meteor & BLEU & SacreBLEU & chRf & TER & BertScore \\\\\n     \\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4}\n     \\cmidrule(lr){5-5}\\cmidrule(lr){6-6}\\cmidrule(lr){7-7}\n     \\cmidrule(lr){8-8}\\cmidrule(lr){9-9}\\cmidrule(lr){10-10}\n     \\cmidrule(lr){11-11}\\cmidrule(lr){12-12}\\cmidrule(lr){13-13}\n     \\cmidrule(lr){14-14}\n     \\\\\n     \\cmidrule{1-14}\n     \n     \n      \\multirow{18}{*}{IWSLT2017} & Zero-Shot & T5 & & 0.619 & {0.386} & 0.588 & 0.587 & 0.445 & {0.254} & {0.308} & 0.577 & 0.573 & 0.870 \\\\\n     \\cmidrule{2-14}\n     & PPO & T5 & SacreBLEU & 0.621 & 0.383 & 0.587 & 0.587 & 0.448 & 0.243 & 0.296 & 0.575 & 0.583 & 0.869\\\\\n     &  & T5 & chRF & 0.622 & 0.385 & 0.590 & 0.590 & {0.448} & 0.248 & 0.301 & {0.578} & 0.575 & {0.870}\\\\\n     &  & T5 & TER & {0.623} & 0.384 & {0.591} & {0.591} & 0.443 & 0.246 & 0.303 & 0.572 & {0.568} & 0.869\\\\\n     &  & T5 & BertScore & 0.533 & 0.326 & 0.507 & 0.507 & 0.321 & 0.143 & 0.174 & 0.406 & 0.573 & 0.839\\\\\n     \\cmidrule{2-14}\n     & NLPO & T5 & SacreBLEU & 0.624 & 0.385 & 0.59 & 0.59 & 0.45 & 0.245 & 0.299 & 0.578 & 0.578 & 0.87 \\\\\n     &  & T5 & chRF & 0.624 & 0.386 & 0.59 & 0.59 & 0.451 & 0.248 & 0.302 & 0.581 & 0.576 & 0.87\\\\\n     &  & T5 & TER & 0.622 & 0.384 & 0.59 & 0.59 & 0.443 & 0.246 & 0.303 & 0.573 & 0.57 & 0.869\\\\\n     &  & T5 & BertScore & 0.611 & 0.377 & 0.58 & 0.58 & 0.425 & 0.239 & 0.291 & 0.555 & 0.573 & 0.866\\\\\n     \\cmidrule{2-14}\n     & Supervised & T5 & & 0.638 & 0.400 & 0.610 & 0.609 & 0.461 & {0.280} & {0.337} & 0.593 & 0.538 & \\textbf{0.878} \\\\\n     \\cmidrule{2-14}\n     & Supervised + PPO & T5 & SacreBLEU & {0.640} & {0.407} & {0.610} & {0.610} & {0.465} & 0.277 & 0.332 & {0.596} & 0.542 & 0.877\\\\\n     &  & T5 & chRF & 0.639 & 0.406 & 0.609 & 0.609 & 0.464 & 0.277 & 0.331 & 0.596 & {0.543} & 0.877\\\\\n     &  & T5 & TER &  0.637 & 0.406 & 0.609 & 0.609 & 0.457 & 0.274 & 0.331 & 0.589 & 0.535 & 0.876\\\\\n     &  & T5 & BertScore & 0.612 & 0.381 & 0.585 & 0.585 & 0.418 & 0.240 & 0.291 & 0.548 & 0.559 & 0.867\\\\\n     \\cmidrule{2-14}\n     & Supervised + NLPO & T5 & SacreBLEU & \\textbf{0.641} & 0.418 & 0.614 & 0.614 & \\textbf{0.474} & 0.289 & 0.343 & \\textbf{0.597} & {0.535} & 0.877\\\\\n     &  & T5 & chRF & 0.643 & 0.418 & 0.621 & 0.621 & 0.464 & \\textbf{0.291} & 0.345 & 0.596 & 0.539 & 0.877\\\\\n     &  & T5 & TER & 0.639 & \\textbf{0.419} & \\textbf{0.621} & \\textbf{0.621} & 0.471 & 0.289 & \\textbf{0.346} & 0.593 & \\textbf{0.535} & 0.877\\\\\n     &  & T5 & BertScore &  0.633 & 0.401 & 0.606 & 0.606 & 0.448 & 0.267 & 0.323 & 0.580 & 0.537 & 0.875\\\\\n     \\bottomrule\n   \\end{tabular}\n   }\n    \\caption{\\textbf{IWSLT test evaluation - lexical and semantic}: Table shows lexical, semantic metrics for RL algorithms with different reward functions bench-marked against supervised baseline models}\n   \\label{tbl:met_lexical_scores}\n\\end{table*}\n\n\\begin{table*}[h]\n   \\centering\n   \\resizebox{1.0\\textwidth}{!}{\n   \\begin{tabular}{@{}cccc|cccccccc@{}}\n   \\toprule\n     Tasks \n     & \\multicolumn{3}{c}{} \n     & \\multicolumn{8}{c}{\\textbf{Diversity Metrics}}\n     \\\\\n     & Alg & Reward Function & LM & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ & Mean Output Length\n     \\\\\n     \\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4}\n     \\cmidrule(lr){5-5}\\cmidrule(lr){6-6}\\cmidrule(lr){7-7}\n     \\cmidrule(lr){8-8}\\cmidrule(lr){9-9}\\cmidrule(lr){10-10}\n     \\cmidrule(lr){11-11}\\cmidrule(lr){12-12}\n     \\\\\n     \n      \\multirow{18}{*}{IWSLT2017} & Zero-Shot & T5 & & 0.662 & 0.097 & 0.4700 & 9.276 & 14.526 & 8312 & 52947 & 18.739 \\\\\n     \\cmidrule{2-12}\n     & PPO & T5 & SacreBLEU & 0.657 & 0.095 & 0.464 & 9.230 & 14.498 & 8285 & 53000 & 19.069\\\\\n     &  & T5 & chRF & 0.660 & 0.096 & 0.468 & 9.253 & 14.526 & 8243 & 53142 & 18.912\\\\\n     &  & T5 & TER & 0.659 & 0.097 & 0.474 & 9.244 & 14.536 & 8129 & 51914 & 18.268\\\\\n     &  & T5 & BertScore & 0.673 & 0.120 & 0.541 & 9.288 & 14.388 & 6642 & 37267 & 11.602\\\\\n     \\cmidrule{2-12}\n     & NLPO & T5 & SacreBLEU & 0.656 & 0.094 & 0.463 & 9.207 & 14.483 & 8240 & 52822 & 19.043\\\\\n     &  & T5 & chRF & 0.658 & 0.095 & 0.464 & 9.233 & 14.502 & 8230 & 53167 & 19.073\\\\\n     &  & T5 & TER & 0.661 & 0.098 & 0.476 & 9.271 & 14.552 & 8223 & 52438 & 18.344\\\\\n     &  & T5 & BertScore &  0.667 & 0.102 & 0.491 & 9.31 & 14.576 & 8134 & 50740 & 17.162\\\\\n     \\cmidrule{2-12}\n     & Supervised & T5 & & 0.655 & 0.095 & 0.467 & 9.210 & 14.492 & 7970 & 51430 & 18.440\\\\\n     \\cmidrule{2-12}\n     & Supervised + PPO & T5 & SacreBLEU &  0.654 & 0.094 & 0.461 & 9.176 & 14.467 & 8061 & 51840 & 18.803\\\\\n     &  & T5 & chRF & 0.656 & 0.094 & 0.464 & 9.202 & 14.497 & 8054 & 52198 & 18.794\\\\\n     &  & T5 & TER & 0.658 &  0.097 & 0.475 & 9.239 & 14.529 & 7969 & 51255 & 18.048\\\\\n     &  & T5 & BertScore &  0.665 & 0.102 & 0.495 & 9.270 & 14.524 & 7495 & 47629 & 16.051\\\\\n     \\cmidrule{2-12}\n     & Supervised + NLPO & T5 & SacreBLEU & 0.655 & 0.094 & 0.461 & 9.182 & 14.469 & 8073 & 51797 & 18.777\\\\\n     &  & T5 & chRF & 0.657 & 0.095 & 0.466 & 9.212 & 14.503 & 8074 & 52056 & 18.642\\\\\n     &  & T5 & TER & 0.658 & 0.096 & 0.473 & 9.231 & 14.521 & 8025 & 51389 & 18.133\\\\\n     &  & T5 & BertScore &0.661 & 0.098 & 0.482 & 9.26 & 14.547 & 7852 & 50672 & 17.496\\\\\n   \\bottomrule\n   \\end{tabular}\n   }\n   \\caption{\\textbf{IWSLT test evaluation - diversity Metrics} Table shows diversity metrics for RL algorithms with different reward functions bench-marked against supervised baseline models}\n   \\label{tbl:mt_diversity_scores}\n\\end{table*}\n\\subsubsection{Results and Discussion}\n\nTables ~\\ref{tbl:met_lexical_scores},~\\ref{tbl:mt_diversity_scores} presents our benchmarking results with 4 reward functions along with supervised baseline performances on test set. Our main finding is that NLPO + Supervised performs better than PPO and supervised models.\n\n\n\\clearpage\n\\subsubsection{Qualitative Results}\n\\label{sec:nmt_qualitative}\nWe show sample generations from each of the algorithms for three randomly picked prompts below.\n\n\\begin{lstlisting}\nSample 1\nPrompt: translate English to German: Several years ago here at TED, Peter Skillman introduced a design challenge  called the marshmallow challenge.\nZero-Shot:  Vor einigen Jahren hat Peter Skillman hier bei TED eine Design-Herausforderung namens Marshmallow Challenge vorgestellt.\nReferences:\nPPO: Vor einigen Jahren, hier bei TED, stellte Peter Skillman einen Design-Wettbewerb namens \"Die Marshmallow-Herausforderung\" vor\nNLPO: Vor einigen Jahren hier bei TED f\u00fchrte Peter Skillman eine Design-Herausforderung namens Marshmallow Challenge ein.\nSupervised: Vor einigen Jahren hier bei TED f\u00fcfchrte Peter Skillman eine Design-Herausforderung namens Marshmallow Challenge ein.\nSupervised + PPO: Vor einigen Jahren hier bei TED, f\u00fchrte Peter Skillman eine Design-Herausforderung namens Marshmallow-Herausforderung ein.\nSupervised + NLPO: Vor einigen Jahren hier bei TED f\u00fcchrte Peter Skillman eine Design-Herausforderung ein, die Marshmallow-Herausforderung.\n\nSample 2\nPrompt: translate English to German: They spend the majority of their time assembling the sticks into ever-growing structures.\nZero-Shot: Sie verbringen den gr\u00f6\u00dften Teil ihrer Zeit damit, die St\u00f6cke in immer wachsende Strukturen zu montieren.\nReferences: Sie investieren einen Gro\u00dfteil ihrer Zeit in den Zusammenbau st\u00e4ndig wachsender Strukturen\nPPO: Sie verbringen den Gro\u00dfteil ihrer Zeit damit, die St\u00f6cke in immer gr\u00f6\u00dfer werdende Strukturen zu montieren.\nNLPO: Sie verbringen den Gro\u00dfteil ihrer Zeit, die St\u00f6cke in immer wachsende Strukturen zu montieren.\nSupervised: Sie verbringen den Gro\u00dfteil ihrer Zeit damit, die St\u00f6cke in immer wachsende Strukturen zu montieren.\nSupervised + PPO: Sie verbringen den Gro\u00dfteil ihrer Zeit damit, die St\u00f6cke in immer wachsende Strukturen zu montieren.\nSupervised + NLPO: Sie verbringen den Gro\u00dfteil ihrer Zeit damit, die St\u00f6cke in immer wachsende Strukturen zu montieren.\n\nSample 3\nPrompt: translate English to German: And of course there are teams that have a lot more \"ta-da\" structures, and among the best are recent graduates of kindergarten.\nZero-Shot: Und nat\u00fcrlich gibt es Teams, die viel mehr \"ta-da\" Strukturen haben, und zu den besten geh\u00f6ren j\u00fcngste Absolventen des Kindergartens.\nReferences: Und nat\u00fcrlich gibt es auch Teams die viel mehr \"ta-da\"-Gebilde haben und unter den Besten sind frische Absolventen des Kindergartens.\nPPO: Und nat\u00fcrlich gibt es Teams, die viel mehr \"ta-da\" Strukturen, und unter den besten sind j\u00fcngste Absolventen des Kindergartens.\nNLPO: Und nat\u00fcrlich gibt es Teams, die viel mehr \"t\na-da\" Strukturen haben, und unter den besten sind j\u00fcngste Absolventen des Kindergartens\nSupervised: Und nat\u00fcrlich gibt es Teams, die viel mehr \"ta-da\"-Strukturen haben, und unter den besten sind j\u00fcngste Absolventen des Kindergartens.\nSupervised + PPO: Und nat\u00fcrlich gibt es Teams, die viel mehr \"ta-da\"-Strukturen haben, und unter den besten sind j\u00fcngste Absolventen des Kindergartens.\nSupervised + NLPO: Und nat\u00fclich gibt es Teams, die viel mehr \"ta-da\"-Strukturen haben, und unter den besten sind j\u00fcngste Absolventen des Kindergartens.\n\\end{lstlisting}\n\n\n\\subsubsection{Setup}\n\nNarrativeQA~\\citep{kovcisky2018narrativeqa} deals with task of generating answers to questions about a given story. For training RL methods, we consider 2 traditional lexical rewards namely Rouge Combined and Rouge-L-Max. We chose T5-base as the base LM since it has been shown to do well at question answering in prior work~\\citep{khashabi-etal-2020-unifiedqa}. We note that the supervised models we use are trained on the UnifiedQA dataset, which contains other QA datasets, and is shown by \\citet{khashabi-etal-2020-unifiedqa} to outperform supervised fine-tuning only on NarrativeQA. Hyperparams for our models can be found in Table~\\ref{tbl:narqa_hyperparams}.\n\n\\begin{table*}[ht!]\n\\centering\n\\footnotesize\n\n\\resizebox{0.5\\textwidth}{!}{\n\\begin{tabular}{ll}\n\\toprule\n\\textbf{Model Params}\n& \\multicolumn{1}{c}{\\textbf{value}} \\\\ \n\\cmidrule{1-2}\nppo\/\\textcolor{forestgreen2}{nlpo} & steps per update: $5120$\\\\\n    & total number of steps: $512000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$\\\\\n    & learning rate: $0.000002$ \\\\\n    & entropy coefficient: $0.0$ \\\\\n    & initial kl coeff: $0.001$ \\\\\n    & target kl: $1.0$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & rollouts top k : $50$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.9$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: $20$} \\\\\n\\cmidrule{1-2}\nsupervised+ ppo (or \\textcolor{forestgreen2}{nlpo}) & steps per update:$2560$\\\\\n    & total number of steps: $512000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$\\\\\n    & learning rate: $0.0000005$ \\\\\n    & entropy coefficient: $0.0$ \\\\\n    & initial kl coeff: $0.001$ \\\\\n    & target kl: $0.2$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & rollouts top k : $50$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.9$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: $20$} \\\\\n\\cmidrule{1-2}\n\\cmidrule{1-2}\ndecoding & num beams: $4$ \\\\\n& max new tokens: $50$\\\\\n\n\\cmidrule{1-2}\ntokenizer & padding side: left\\\\\n& truncation side: right\\\\\n& max length: 512 \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{NarQA Hyperparams}: Table shows a list of all hyper-parameters and their settings}\n       \\label{tbl:narqa_hyperparams}\n\\end{table*}\n\n\n\\begin{landscape}\n\\begin{table*}[h]\n   \\centering\n   \\resizebox{1.5\\textwidth}{!}{\n   \\begin{tabular}{@{}cccc|cccccccc|cccccccc@{}}\n   \\toprule\n     Tasks \n     & \\multicolumn{3}{c}{} \n     & \\multicolumn{8}{c}{\\textbf{Lexical and Semantic Metrics}} \n     & \\multicolumn{8}{c}{\\textbf{Diversity Metrics}}\n     \\\\\n     & Alg & Reward Function & LM & Rouge-1 & Rouge-2 & Rouge-L & Rouge-LSum & Rouge-LMax & Meteor & BLEU & BertScore & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ & Mean Output Length\n     \\\\\n     \\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-12} \\cmidrule(lr){13-20}\n     \\\\\n     \\multirow{3}{*}{NarQA} & \n     \\\\\n     \\cmidrule(lr){2-20}\n     & Zero Shot &  & T5 & 0.095 & 0.022 & 0.084 & 0.084 & 0.117 & 0.095 & 0.009 & 0.835 & 0.415 & 0.026 & 0.097 & 9.641 & 13.468 & 1880 & 11495 & 31.688\\\\\n    \\cmidrule(lr){2-20} \n     & PPO & Rouge Combined & T5 & 0.101 &0.025 &0.088 &0.088 &0.122 &0.099 &0.01 &0.837 &0.462 &0.03 &0.125 &9.759 &13.789 &2522 &17806 &32.352 \\\\\n     &  & Rouge-L Max & T5 & 0.099 &0.025 &0.087 &0.087 &0.122 &0.099 &0.01 &0.835 &0.439 &0.029 &0.119 &9.653 &13.618 &2292 &15816 &31.479 \\\\\n     \\cmidrule(lr){2-20} \n     & NLPO & Rouge Combined & T5 & 0.097 &0.023 &0.085 &0.085 &0.118 &0.098 &0.009 &0.836 &0.418 &0.025 &0.096 &9.652 &13.528 &1816 &10980 &32.117\\\\\n     & & Rouge-L Max & T5 & 0.102 &0.026 &0.089 &0.089 &0.124 &0.1 &0.01 &0.837 &0.445 &0.029 &0.118 &9.776 &13.75 &2181 &14569 &31.555 \\\\\n     \n     \\cmidrule(lr){2-20}\n     & Supervised &  & T5 & 0.378 & 0.190 & 0.367 & 0.367 & 0.581 & 0.099 & 0.209 & 0.931 & 0.609 & 0.156 & 0.534 &  9.807 & 13.657 & 3250 & 14995 & 4.923\n     \\\\\n        \n     \\cmidrule(lr){2-20} \n     & Supervised + PPO & Rouge Combined & T5 & 0.38 &0.177 &0.371 &0.371 &0.585 &0.09 &0.229 &0.931 &0.64 &0.174 &0.559 &10.132 &13.547 &3326 &13785 &4.353 \\\\\n     &  & Rouge-L Max & T5 & 0.368 &0.18 &0.36 &0.36 &0.585 &0.083 &0.239 &0.931 &0.641 &0.187 &0.576 &10.201 &13.452 &3287 &12436 &3.913 \\\\\n     \\cmidrule(lr){2-20} \n     & Supervised + NLPO & Rouge Combined & T5 & 0.398 &0.21 &0.393 &0.373 &0.589 &0.096 &0.24 &0.971 &0.679 &0.185 &0.595 &10.304 &13.694 &3371 &15067 &4.728 \\\\\n     & & Rouge-L Max & T5 & 0.381 &0.194 &0.383 &0.383 &0.588 &0.093 &0.243 &0.932 &0.645 &0.187 &0.59 &10.2 &13.397 &3287 &12171 &3.889\n\n     \\\\\n   \\bottomrule\n   \\end{tabular}\n   }\n   \\caption{\\textbf{Evaluation of NarrativeQA}: Reference Metrics, supervised is based on UnifiedQA~\\citep{khashabi-etal-2020-unifiedqa}.}\n   \\label{tbl:narrativeqa_lexical_scores}\n\\end{table*}\n\\end{landscape}\n\\subsubsection{Results and Discussion}\nTable ~\\ref{tbl:narrativeqa_lexical_scores} presents our benchmarking results with 2 reward functions along with supervised baseline performances on the NarrativeQA test set. Similar to other methods, our main finding is that warm-started initial policies are crucial for learning to generate answers that successfully use the input context.\n\n\\subsubsection{Qualitative Results}\n\\label{sec_nmt_qualitative}\nWe show sample generations from each of the algorithms for three randomly picked prompts below.\n\n\\begin{lstlisting}\nSample 1\nPrompt: who is mark hunter? mark hunter (slater), a high school student in a sleepy suburb of phoenix, arizona, starts an fm pirate  radio station that broadcasts from the basement of his parents' house. mark is a loner, an outsider, whose only outlet for his teenage angst and aggression is his unauthorized radio station. his pirate station's theme song is \"everybody knows\" by leonard cohen and there are glimpses of cassettes by such alternative musicians as the jesus and mary chain, camper van beethoven, primal scream, soundgarden, ice-t, bad brains, concrete blonde, henry rollins, and the pixies. by day, mark is seen as a loner, hardly talking to anyone around him; by night, he expresses his outsider views about what is wrong with american society. when he speaks his mind about what is going on at his school and in the community, more and more of his fellow students tune in to hear his show.nobody knows the true identity of \"hard harry\" or \"happy harry hard-on,\" as mark  refers to himself, until nora diniro (mathis), a fellow student, tracks him down and confronts him the day after a student named malcolm commits  suicide after harry attempts to reason with him. the radio show becomes increasingly popular and influential after harry confronts the suicide head-on, exhorting his listeners to do something about their problems instead of surrendering to them through suicide\u2014at the crescendo of his yelled speech, an overachieving student named paige woodward (who has been a constant listener) jams her various medals and accolades into a microwave and turns it on. she then sits, watching the awards cook until the microwave explodes, injuring her. while this is happening, other students act out in cathartic release.eventually, the radio show causes so much trouble in the community that the fcc is called in to investigate. during the fracas, it is revealed that the school's principal (annie ross) has been expelling \"problem students,\" namely, students with below-average standardized test scores, in an effort to boost the district's test scores while still keeping their names on the rolls (a criminal offense) in order to retain government funding.realizing he has started something huge, mark decides it is up to him to end it. he dismantles his radio station and attaches it to his mother's old jeep, creating a mobile transmitter so his position can't be triangulated. pursued by the police and the fcc, nora drives the jeep around while mark broadcasts. the harmonizer he uses to disguise his voice breaks, and with no time left to fix it, mark decides to broadcast his final message as himself. they finally drive up to the crowd of protesting students, and mark tells them that the world belongs to them and that they should make their own future. the police step in and arrest mark and nora. as they are taken away, mark reminds the students to \"talk hard.\" as the film ends, the voices of other students (and even one of the teachers) speak as intros for their own independent stations, which can be heard broadcasting across the country.\nZero-Shot: hunter (slater) is a high school student in a sleepy suburb of phoenix, arizona. he starts an fm pirate radio station that broadcasts from the basement of his\nReferences: he is a high school student in phoenix 2) loner and outsider student with a radio station\nPPO: hunter (slater), a high school student in a sleepy suburb of phoenix, arizona, starts an fm pirate radio station that broadcasts from the basement of his parents' house\nNLPO: hunter (slater), a high school student in a sleepy suburb of phoenix, arizona, starts an fm pirate radio station that broadcasts from the basement of his parents' house\nSupervised: a high school student\nSupervised + PPO: a high school student\nSupervised + NLPO: a high school student\n\nSample 2\nPrompt: what was the principle doing with the problem students? mark hunter (slater), a high school student in a sleepy suburb of phoenix, arizona, starts an fm pirate radio station that broadcasts from the basement of his parents' house. mark is a loner, an outsider, whose only outlet for his teenage angst and aggression is his unauthorized radio station. his pirate station's theme song is \"everybody knows\" by leonard cohen and there are glimpses of cassettes by such alternative musicians as the jesus and mary chain, camper van beethoven, primal scream, soundgarden, ice-t, bad brains, concrete blonde, henry rollins, and the pixies. by day, mark is seen as a loner, hardly talking to anyone around him; by night, he expresses his outsider views about what is wrong with american society. when he speaks his mind about what is going on at his school and in the community, more and more of his fellow students tune in to hear his show.nobody knows the true identity of \"hard harry\" or \"happy harry hard-on,\" as mark refers to himself, until nora diniro (mathis), a fellow student, tracks him down and confronts him the day after a student named malcolm commits suicide after harry attempts to reason with him. the radio show becomes increasingly popular and influential after harry confronts the suicide head-on, exhorting his listeners to do something about their problems instead of surrendering to them through suicide\u20ac\"at the crescendo of his yelled speech, an overachieving student named paige woodward (who has been a constant listener) jams her various medals and accolades into a microwave and turns it on. she then sits, watching the awards cook until the microwave explodes, injuring her. while this is happening, other students act out in cathartic release.eventually, the radio show causes so much trouble in the community that the fcc is called in to investigate. during the fracas, it is revealed that the school's principal (annie ross) has been expelling \"problem students,\" namely, students with below-average standardized test scores, in an effort to boost the district's  test scores while still keeping their names on the rolls (a criminal offense) in order to retain government funding.realizing he has started  something huge, mark decides it is up to him to end it. he dismantles his radio station and attaches it to his mother's old jeep, creating a  mobile transmitter so his position can't be triangulated. pursued by the police and the fcc, nora drives the jeep around while mark broadcasts.  the harmonizer he uses to disguise his voice breaks, and with no time left to fix it, mark decides to broadcast his final message as himself.  they finally drive up to the crowd of protesting students, and mark tells them that the world belongs to them and that they should make their own future. the police step in and arrest mark and nora. as they are taken away, mark reminds the students to \"talk hard.\" as the film ends, the voices of other students (and even one of the teachers) speak as intros for their own independent stations, which can be heard broadcasting  across the country.\nZero-Shot: mark hunter (slater) starts an fm pirate radio station that broadcasts from the basement of his parents' house. by day, mark is seen as a loner, an outsider, whose only outlet\nReferences: 1) expelling them. 2) expelling the students\nPPO: mark hunter (slater) starts an fm pirate radio station that broadcasts from the basement of his parents' house. mark is a loner, an outsider, whose only outlet for his teenage ang \nNLPO: mark hunter (slater) starts an fm pirate radio station that broadcasts from the basement of his parents' house. mark is a loner, an outsider, whose only outlet for his teenage ang\nSupervised: he was taking them to a psychiatric hospital.\nSupervised + PPO: addressing them \nSupervised + NLPO: he was expelling them\n\nSample 3\nPrompt: what happens to the people maskull meets in his travels to various lands? maskull,  a man longing for adventures, accepts an invitation from krag, an acquaintance of his friend nightspore, to travel to tormance after a seance.  the three set off in a crystal ship from an abandoned observatory in scotland but maskull awakens to find himself alone on tormance. in every  land he passes through he usually meets only one or two persons; these meetings often (though not always) end in the death of those he meets,  either at his own hand or by that of another. he learns of his own impending death, meets krag again, and dies shortly after learning that he is in  fact nightspore himself. the book concludes with a final revelation from krag (who claims to be known on earth as \"pain\") to nightspore about  the origin of the universe. the author turns out to support a variation of the doctrine of the demiurge, somewhat similar to that propounded  by some gnostics.all of the characters and lands are types used to convey the author's critique of several philosophical systems. on tormance,  most such viewpoints or ways of life are accompanied by corresponding new bodily sense organs or modifications of the same, thus each distinct  weltanschauung landscape has its corresponding sensorium.\nZero-Shot: what happens to the people maskull meets in his travels to various lands? n maskull, a man longing for adventures, accepts an invitation from krag to travel to tormance after \nReferences: 1) they usually die by his hand or anothers. 2) they die\nPPO: what happens to the people maskull meets in his travels to various lands? n maskull, a man longing for adventures, accepts an invitation from krag to travel to tormance after\nNLPO:  maskull meets krag, an acquaintance of nightspore, to travel to tormance after a seance. maskull awakens to find himself alone on tormance. in every land \nSupervised: they die either from his own hand or from another person's death. \nSupervised + PPO: they end up dying either at his own hand or by another's. \nSupervised + NLPO: they end up dying either at his own hand or by another's\n\\end{lstlisting}\n\\subsubsection{Setup}\n\nCommonGen \\citep{lin-etal-2020-commongen} deals with task of generating coherent sentences describing an input set of concepts (eg. \"a man is throwing a frisbee\"). For training RL methods, we consider 3 traditional lexical rewards namely Rouge-1, Rouge-avg (which is an average of Rouge-1, 2 and L) and meteor. Additionally, we also train with task-specific rewards such as CIDEr~ \\citep{vedantam2015cider}, SPICE~\\citep{anderson2016spice} and SPiDer~\\citep{liu2017improved}  which is a just a linear combination of both with equal weights. We chose T5-base as the base LM since it is well-suited for structure to text tasks. We additionally note that concept set inputs are prefixed with \"generate a sentence with:\" to encourage exploration. \n\nDuring our initial experiments when fine-tuning directly on LM, we observed that policy learns to repeat the prompted concepts in order to maximize rewards resulting in a well-known problem of \\textit{reward hacking}. To mitigate this, we add a penalty score of $-1$ to final task reward if the n-grams of prompt text overlaps with generated text. In contrast, when initialized with a supervised policy, this problem is not seen and hence penalty score is not applied. We use beam search as the decoding method during evaluation whereas for rollouts, we use top k sampling to favor exploration over exploitation. Table~\\ref{tbl:common_gen_hyperparams} provides an in-depth summary of setting of hyperparameter values along with other implementation details.\n\n\\begin{table*}[ht!]\n\\centering\n\\footnotesize\n\n\\resizebox{0.5\\textwidth}{!}{\n\\begin{tabular}{ll}\n\\toprule\n\\textbf{Model Params}\n& \\multicolumn{1}{c}{\\textbf{value}} \\\\ \n\\cmidrule{1-2}\nsupervised & batch size: $8$\\\\\n& epochs: $4$ \\\\\n& learning rate: $0.00001$ \\\\\n& learning rate scheduler: cosine \\\\\n& weight decay: $0.01$ \\\\\n\\cmidrule{1-2}\nppo\/ \\textcolor{forestgreen2}{nlpo} & steps per update: $1280$\\\\\n    & total number of steps: $256000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$ \\\\\n    & learning rate: $0.000002$ \\\\\n    & entropy coefficient: $0.01$ \\\\\n    & initial kl coeff: $0.001$ \\\\\n    & target kl: $2.0$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.9$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: $20$} \\\\\n\\cmidrule{1-2}\nsupervised+ ppo (or \\textcolor{forestgreen2}{nlpo}) & steps per update: $1280$\\\\\n    & total number of steps: $128000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$ \\\\\n    & learning rate: $0.000002$ \\\\\n    & entropy coefficient: $0.01$ \\\\\n    & initial kl coeff: $0.01$ \\\\\n    & target kl: $1.0$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.9$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: $20$} \\\\\n\\cmidrule{1-2}\n\\cmidrule{1-2}\ndecoding & num beams: $5$ \\\\\n& min length: $5$ \\\\\n& max new tokens: $20$\\\\\n\n\\cmidrule{1-2}\ntokenizer & padding side: left\\\\\n& max length: $20$ \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{CommonGen Hyperparams}: Table shows a list of all hyper-parameters and their settings}\n       \\label{tbl:common_gen_hyperparams}\n\\end{table*}\n\n\\subsubsection{Results and Discussion}\n\nTables ~\\ref{tbl:common_gen_dev_scores},~\\ref{tbl:common_gen_test_scores} presents our benchmarking results with 6 reward functions along with supervised baseline performances on dev and test sets respectively. Our main finding is that warm-started initial policies are crucial for learning to generate coherent sentences with common sense. Without warm-start, policies suffer from reward hacking despite application of repetition penalty and task-specific metrics such as CIDer etc. Further, we find that RL fine-tuned models obtain very high concept coverage which is also seen in Table~\\ref{app:tbl_common_gen_qualitative}.\nSupervised models often tend to miss few concepts in its generation compared to RL methods. \n\n\n\\begin{table*}[h]\n   \\centering\n   \\resizebox{1.0\\textwidth}{!}{\n   \\begin{tabular}{@{}ccccccccccccc@{}}\n   \\toprule\n     Tasks \n     & \\multicolumn{3}{c}{\\textbf{\\_}} \n     & \\multicolumn{6}{c}{\\textbf{Lexical and Semantic Metrics}}\n     \\\\\n     & Alg & LM & Reward function & Rouge-2\t& Rouge-L & Bleu (n=3) & \tBleu (n=4) & Meteor & CIDEr & SPICE & Coverage\\\\\n     \\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4}\n     \\cmidrule(lr){5-5} \\cmidrule(lr){6-6} \\cmidrule(lr){7-7}\n     \\cmidrule(lr){8-8} \\cmidrule(lr){9-9} \\cmidrule(lr){10-10}\n     \\cmidrule(lr){11-11} \\cmidrule(lr){12-12}\n      \\\\\n      \\multirow{22}{*}{CommonGen} & Zero-Shot & T5 & & 0.016 & 0.264 & 0.029 & 0.006 & 0.203 & 6.200 & 0.115 & 91.070\\\\\n       \\cmidrule{2-12}\n      & PPO & T5 & Rouge-1 & 0.085 $\\pm$ 0.008 &\t0.354 $\\pm$ 0.004 &\t0.161 $\\pm$ 0.011 & 0.087 $\\pm$ 0.009 &\t0.235 $\\pm$ 0.002 &\t8.673 $\\pm$ 0.234 & 0.157 $\\pm$ 0.001 & 88.544 $\\pm$ 2.36\\\\\n       & & T5 & Rouge-Avg & 0.093 $\\pm$ 0.005 & 0.351 $\\pm$ 0.001 & 0.169 $\\pm$ 0.032 &\t0.097 $\\pm$ 0.017 & 0.224 $\\pm$ 0.012 & 8.212 $\\pm$ 1.329 & 0.159 $\\pm$ 0.011 & 82.584 $\\pm$ 2.569\n       \\\\\n       & & T5 & Meteor & 0.091 $\\pm$ 0.008 &\t0.308 $\\pm$ 0.007 &\t0.166 $\\pm$ 0.016 & 0.088 $\\pm$ 0.013 &\t0.220 $\\pm$ 0.006 &\t7.251 $\\pm$ 0.453\t& 0.161 $\\pm$ 0.007 & 79.718 $\\pm$ 2.267\n       \\\\\n       & & T5 & SPice & 0.065 $\\pm$ 0.003\t& 0.302 $\\pm$ 0.002 & \t0.115 $\\pm$ 0.063 & 0.067 $\\pm$ 0.041 & 0.193 $\\pm$ 0.014 & 6.571 $\\pm$ 1.312 &\t0.175 $\\pm$ 0.011 & 69.340 $\\pm$ 3.617\n       \\\\\n       & & T5 & CiDer & 0.066 $\\pm$ 0.003\t& 0.304 $\\pm$ 0.002 &\t0.132 $\\pm$ 0.057 & 0.074 $\\pm$ 0.036 &\t0.211 $\\pm$ 0.009 &\t6.877 $\\pm$ 1.218 & 0.143 $\\pm$ 0.017 &\t80.114 $\\pm$ 4.852\n       \\\\\n       & & T5 & SPider &  0.117 $\\pm$ 0.005 & 0.352 $\\pm$ 0.007 &\t0.224 $\\pm$ 0.014\t& 0.137 $\\pm$ 0.011 &\t0.226 $\\pm$ 0.01 &\t9.162 $\\pm$ 0.539 & 0.186 $\\pm$ 0.006 &\t73.374 $\\pm$ 6.073\n       \\\\ \n       \\cmidrule(lr){2-12}\n       & NLPO & T5 & Rouge-1 & 0.087 $\\pm$ 0.002 & 0.339 $\\pm$ 0.009 & 0.127 $\\pm$ 0.048\t & 0.069 $\\pm$ 0.035 &\t0.213 $\\pm$ 0.002 &\t6.962 $\\pm$ 0.883 &\t0.145 $\\pm$ 0.022 &80.89 $\\pm$ 9.544\\\\\n        & & T5 & Rouge-Avg & 0.095 $\\pm$ 0.001 &0.338 $\\pm$ 0.002 & 0.159 $\\pm$ 0.02 &\t0.093 $\\pm$ 0.013 & 0.216 $\\pm$ 0.009 &  7.55 $\\pm$ 0.688\t& 0.153 $\\pm$ 0.008 & 77.944 $\\pm$ 2.770\n       \\\\\n       & & T5 & Meteor & 0.110 $\\pm$ 0.005 & 0.332 $\\pm$ 0.003 &\t0.214 $\\pm$ 0.007 &\t0.124 $\\pm$ 0.007 & \t0.235 $\\pm$ 0.004 & 8.669 $\\pm$ 0.164 &\t0.173 $\\pm$ 0.002 &\t82.007 $\\pm$ 1.012\n       \\\\\n       & & T5 & SPice & 0.014 $\\pm$ 0.006 & 0.242 $\\pm$ 0.001 & 0.037 $\\pm$ 0.011 & 0.018 $\\pm$ 0.007 & 0.156 $\\pm$ 0.007 & 4.685 $\\pm$ 0.283 & 0.168 $\\pm$ 0.008 &\t56.998 $\\pm$ 3.548 \n       \\\\\n       & & T5 & CiDer & 0.046 $\\pm$ 0.001 &\t0.241 $\\pm$ 0.003 & 0.078 $\\pm$ 0.028 & 0.043 $\\pm$ 0.016 & 0.143 $\\pm$ 0.018  & 3.964 $\\pm$ 0.792 & 0.103 $\\pm$ 0.012 &\t49.606 $\\pm$ 7.971\n       \\\\\n       & & T5 & SPider &  0.060 $\\pm$ 0.006 & 0.258 $\\pm$ 0.001 &\t0.090 $\\pm$ 0.008 & \t0.056 $\\pm$ 0.005 &\t0.151 $\\pm$ 0.022 &\t4.411 $\\pm$ 0.837 &\t0.123 $\\pm$ 0.022 &\t49.230 $\\pm$ 10.468\n       \\\\  \n       \\cmidrule(lr){2-12}\n      & Supervised & T5 & & 0.215 $\\pm$ 0.001 & 0.438 $\\pm$  0.001 & 0.444 $\\pm$ 0.001 & 0.329 $\\pm$ 0.001 &\t0.321 $\\pm$ 0.001\t& 16.385 $\\pm$ 0.046 & \t\\textbf{0.299} $\\pm$ 0.001 & 94.476 $\\pm$ 0.172 \\\\\n      \\cmidrule(lr){2-12}\n      & Supervised + PPO & T5 & Rouge-1 & 0.232 $\\pm$ 0.002 & \\textbf{0.453} $\\pm$ 0.002 & 0.454 $\\pm$ 0.006 &\t\\textbf{0.338} $\\pm$ 0.006 & 0.320 $\\pm$ 0.002 &\t16.233 $\\pm$ 0.159 &\t0.288 $\\pm$ 0.004 & \t96.412 $\\pm$ 0.424\\\\\n       & & T5 & Rouge-Avg & 0.230 $\\pm$ 0.001 & 0.450 $\\pm$ 0.001 &\t0.448 $\\pm$ 0.005 & 0.334 $\\pm$ 0.005 &\t0.319 $\\pm$ 0.001 & 16.069 $\\pm$ 0.167 &\t0.287 $\\pm$ 0.003 & 96.116 $\\pm$ 0.679\n       \\\\\n       & & T5 & Meteor & \\textbf{0.234} $\\pm$ 0.002 & 0.450 $\\pm$ 0.003 &\\textbf{0.462} $\\pm$ 0.007 & 0.342 $\\pm$ 0.007 & \\textbf{0.327} $\\pm$ 0.001 & \\textbf{16.797} $\\pm$ 0.152 & 0.295 $\\pm$ 0.001 &\t\\textbf{97.690} $\\pm$ 0.371\n       \\\\\n       & & T5 & SPice & 0.227 $\\pm$ 0.004 &\t0.447 $\\pm$ 0.003 & 0.450 $\\pm$ 0.007 & 0.336 $\\pm$ 0.008 & 0.319 $\\pm$ 0.002 &\t16.208 $\\pm$ 0.249 & 0.288 $\\pm$ 0.003 & \t96.492 $\\pm$ 0.29 \n       \\\\\n       & & T5 & CiDer & 0.224 $\\pm$ 0.003 &\t0.446 $\\pm$ 0.003 & 0.427 $\\pm$ 0.012 & 0.309 $\\pm$ 0.01 & 0.316 $\\pm$ 0.004 &\t15.497 $\\pm$ 0.428 & 0.283 $\\pm$ 0.004 & \t96.344 $\\pm$ 0.547\n       \\\\\n       & & T5 & SPider &  0.226 $\\pm$ 0.003 & 0.448 $\\pm$ 0.002 &\t0.436 $\\pm$ 0.005 &\t0.319 $\\pm$ 0.004 &\t0.317 $\\pm$ 0.003 &\t15.678 $\\pm$ 0.192 & 0.281 $\\pm$ 0.003 &\t96.154 $\\pm$ 0.426\n       \\\\ \n       \\cmidrule(lr){2-12}\n       \n       & Supervised + NLPO & T5 & Rouge-1 & 0.229 $\\pm$ 0.002 &\t0.450 $\\pm$ 0.001 &\t0.454 $\\pm$ 0.005 &\t0.338 $\\pm$ 0.004 &\t0.320 $\\pm$ 0.003 &\t16.206 $\\pm$ 0.175 &\t0.289 $\\pm$ 0.002 &\t96.342 $\\pm$ 0.572\\\\\n       & & T5 & Rouge-Avg & 0.232 $\\pm$ 0.003 &\t0.451 $\\pm$ 0.002 &\t0.458 $\\pm$ 0.01 &\t0.342 $\\pm$ 0.009 &\t0.321 $\\pm$ 0.003 &  16.351 $\\pm$ 0.335 &\t0.290 $\\pm$ 0.005 & 95.998 $\\pm$ 0.496\n       \\\\\n       & & T5 & Meteor & 0.231 $\\pm$ 0.003 & 0.449 $\\pm$ 0.002 &\t0.454 $\\pm$ 0.007 &\t0.334 $\\pm$ 0.008 &\t0.326 $\\pm$ 0.002 & 16.574 $\\pm$ 0.269 & 0.292 $\\pm$ 0.003 &\t97.374 $\\pm$ 0.457\n       \\\\\n       & & T5 & SPice &  0.223 $\\pm$ 0.002 & 0.442 $\\pm$ 0.001 & 0.435 $\\pm$ 0.011 &\t0.321 $\\pm$ 0.010 &\t0.315 $\\pm$ 0.004 &\t15.747 $\\pm$ 0.401 & 0.283 $\\pm$ 0.005 & 96.25 $\\pm$ 0.313\n       \\\\\n       & & T5 & CiDer & 0.226 $\\pm$ 0.002 &\t0.447 $\\pm$ 0.004 & 0.433 $\\pm$ 0.007 & 0.315 $\\pm$ 0.008 & 0.318 $\\pm$ 0.003 &\t15.741 $\\pm$ 0.170 & 0.285 $\\pm$ 0.001 &\t96.354 $\\pm$ 0.971\n       \\\\\n       & & T5 & SPider &  0.226 $\\pm$ 0.004 & 0.447 $\\pm$ 0.003 &\t0.434 $\\pm$ 0.006 &\t0.316 $\\pm$ 0.006 &\t0.319 $\\pm$ 0.002 & 15.739 $\\pm$ 0.311 & 0.284 $\\pm$ 0.003 &\t96.333 $\\pm$ 0.644\\\\\n       \\bottomrule\n        \n       \\end{tabular}\n       }\n       \\caption{\\textbf{CommonGen test evaluation} Table shows official scores obtained from CommonGen hold-out evaluation. The most important result is that RL fine-tuning on a supervised model yields better performance across most metrics especially Coverage which indicates the ratio of concepts covered in generated texts}\n       \\label{tbl:common_gen_test_scores}\n    \\end{table*}\n\n\n\\begin{landscape}\n  \\begin{table*}[h]\n     \\centering\n     \\resizebox{1.5\\textwidth}{!}{\n     \\begin{tabular}{@{}cccccccccccccccccccccc@{}}\n     \\toprule\n       Tasks \n       & \\multicolumn{4}{c}{\\textbf{\\_}} \n       & \\multicolumn{9}{c}{\\textbf{Lexical and Semantic Metrics}} \n       & \\multicolumn{8}{c}{\\textbf{Diversity Metrics}}\n       \\\\\n       & Alg & Reward Function & Top k & LM & Rouge-1 & Rouge-2 & Rouge-L & Rouge-LSum & Meteor & BLEU & BertScore & Cider & Spice & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ & Mean Output Length\n       \\\\\n       \\cmidrule(lr){2-2}\\cmidrule(lr){3-3} \\cmidrule(lr){4-4} \\cmidrule(lr){5-5} \\cmidrule(lr){6-14} \\cmidrule(lr){15-22}\n       \\\\\n       \\multirow{31}{*}{CommonGen} & Zero-Shot &   &   & T5 & 0.415 & 0.016 & 0.270 & 0.270 & 0.179 & 0.0 & 0.854 & 0.640 & 0.231 & 0.430 & 0.090 & 0.335 & 5.998 & 7.957 & 345 & 1964 & 8.797\n       \\\\\n  \n       \\cmidrule(lr){2-20} \n        \\cmidrule(lr){2-22}\n       & PPO & Rouge-1 & 50 & T5 & 0.537 $\\pm$ 0.004 & 0.093 $\\pm$ 0.012 &  0.380 $\\pm$ 0.006 & 0.380 $\\pm$ 0.006 & 0.235 $\\pm$ 0.005 & 0.016 $\\pm$ 0.002 & 0.896 $\\pm$ 0.001 & 0.950 $\\pm$ 0.015 & 0.318 $\\pm$ 0.016 & 0.526 $\\pm$ 0.020 & 0.128 $\\pm$ 0.005 & 0.518 $\\pm$ 0.036 & 6.679 $\\pm$ 0.132 & 10.572 $\\pm$ 0.234 & 437.4 $\\pm$ 42.017 & 2418.8 $\\pm$ 167.947 & 7.214 $\\pm$ 0.374 \\\\\n       \\cmidrule(lr){3-22}\n       & & Rouge-Avg  & 50 & T5 & 0.519  $\\pm$ 0.0185 & 0.102 $\\pm$ 0.007 &  0.377 $\\pm$ 0.013 & 0.376 $\\pm$ 0.014 & 0.225 $\\pm$ 0.024 & 0.020 $\\pm$ 0.002 & 0.897  $\\pm$ 0.005 & 0.921 $\\pm$ 0.102 & 0.328 $\\pm$ 0.009 & 0.536 $\\pm$ 0.069 & 0.141 $\\pm$ 0.022 & 0.510  $\\pm$ 0.056 & 6.777 $\\pm$ 0.539 & 10.348 $\\pm$ 0.134 & 458.6 $\\pm$ 19.734 & 2244.4 $\\pm$ 162.855 & 6.887  $\\pm$ 1.006\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & Meteor & 50 & T5 & 0.411 $\\pm$ 0.009 &  0.090 $\\pm$ 0.008 & 0.304  $\\pm$ 0.006 & 0.304  $\\pm$ 0.006 &  0.210  $\\pm$ 0.005 & 0.029  $\\pm$ 0.004  & 0.875 $\\pm$ 0.007 & 0.638 $\\pm$ 0.048 & 0.259 $\\pm$ 0.017 & 0.547  $\\pm$ 0.012 &  0.147 $\\pm$ 0.003 & 0.529 $\\pm$ 0.014 & 7.62 $\\pm$ 0.127 & 11.464 $\\pm$ 0.151 & 1039.4 $\\pm$ 63.276 & 5197.2 $\\pm$ 280.004 &  13.660 $\\pm$ 0.324\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & SPice & 50 & T5 & 0.439 $\\pm$ 0.035 & 0.079 $\\pm$ 0.045 & 0.323 $\\pm$ 0.036 & 0.323 $\\pm$ 0.036  & 0.183 $\\pm$ 0.022 & 0.012 $\\pm$ 0.009 & 0.891 $\\pm$ 0.005 & 0.777 $\\pm$ 0.140 & 0.400 $\\pm$ 0.012 & 0.546 $\\pm$ 0.054 &  0.149 $\\pm$ 0.019 & 0.545 $\\pm$ 0.072 & 6.721 $\\pm$ 0.441 & 10.492 $\\pm$ 0.330 & 409.2 $\\pm$ 41.605 & 1878.4 $\\pm$ 167.492 & 5.706 $\\pm$ 0.678\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & CiDer & 50 & T5 & 0.453 $\\pm$ 0.038 & 0.081 $\\pm$ 0.037 &  0.326 $\\pm$ 0.033 & 0.326 $\\pm$ 0.033 & 0.203 $\\pm$ 0.022 &  0.017 $\\pm$ 0.009 & 0.885 $\\pm$ 0.008 & 0.770 $\\pm$ 0.134 &  0.291 $\\pm$ 0.036 & 0.597 $\\pm$ 0.081 & 0.195  $\\pm$ 0.040 & 0.639 $\\pm$ 0.106 & 7.732 $\\pm$ 0.682 & 11.131 $\\pm$ 0.502 & 777.0 $\\pm$ 144.676 & 3350.8 $\\pm$ 503.419 &  7.393 $\\pm$ 0.572\n       \\\\\n       \\cmidrule(lr){3-22}\n        \\rowcolor{lightgray}\n       & & SPider & 50 & T5 & 0.512 $\\pm$ 0.008 & 0.141 $\\pm$ 0.007 & 0.388 $\\pm$ 0.002 & 0.388 $\\pm$ 0.003 & 0.242 $\\pm$ 0.007 & 0.032 $\\pm$ 0.003 & 0.902 $\\pm$ 0.001 & 1.045 $\\pm$ 0.034 & 0.380 $\\pm$ 0.006 & 0.482 $\\pm$ 0.015 & 0.133 $\\pm$ 0.003 & 0.472 $\\pm$ 0.021 & 6.372 $\\pm$ 0.221 & 10.303 $\\pm$ 0.228 & 502.6 $\\pm$ 33.422 & 2281.4 $\\pm$ 252.471 & 7.489 $\\pm$ 0.358\n       \\\\\n      \n        \\cmidrule(lr){2-22}\n       & NLPO & Rouge-1 & 50 & T5 & 0.499 $\\pm$ 0.012 &0.089 $\\pm$ 0.003 &0.328 $\\pm$ 0.007 &0.328 $\\pm$ 0.007 &0.198 $\\pm$ 0.002 &0.021 $\\pm$ 0.001 &0.872 $\\pm$ 0.005 &0.815 $\\pm$ 0.009 &0.305 $\\pm$ 0.008 &0.559 $\\pm$ 0.01 &0.148 $\\pm$ 0.003 &0.555 $\\pm$ 0.012 &7.059 $\\pm$ 0.067 &10.657 $\\pm$ 0.105 &457.9 $\\pm$ 11.108 &2349.6 $\\pm$ 60.345 &6.586 $\\pm$ 0.094\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & Rouge-Avg  & 50 & T5 & 0.47 $\\pm$ 0.01 &0.096 $\\pm$ 0.004 &0.312 $\\pm$ 0.006 &0.312 $\\pm$ 0.006 &0.202 $\\pm$ 0.008 &0.025 $\\pm$ 0.002 &0.843 $\\pm$ 0.013 &0.816 $\\pm$ 0.026 &0.299 $\\pm$ 0.007 &0.512 $\\pm$ 0.019 &0.146 $\\pm$ 0.011 &0.513 $\\pm$ 0.012 &6.781 $\\pm$ 0.15 &10.424 $\\pm$ 0.156 &484.18 $\\pm$ 17.303 &2357.54 $\\pm$ 152.113 &7.131 $\\pm$ 0.487\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & Meteor & 50 & T5 & 0.389 $\\pm$ 0.013 &0.1 $\\pm$ 0.004 &0.293 $\\pm$ 0.008 &0.293 $\\pm$ 0.008 &0.226 $\\pm$ 0.024 &0.035 $\\pm$ 0.004 &0.832 $\\pm$ 0.018 &0.691 $\\pm$ 0.04 &0.266 $\\pm$ 0.016 &0.503 $\\pm$ 0.003 &0.132 $\\pm$ 0.005 &0.471 $\\pm$ 0.008 &7.146 $\\pm$ 0.192 &10.727 $\\pm$ 0.313 &648.05 $\\pm$ 33.963 &3536.0 $\\pm$ 444.638 &11.062 $\\pm$ 1.301\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & SPice & 50 & T5 & 0.329 $\\pm$ 0.015 &0.036 $\\pm$ 0.008 &0.247 $\\pm$ 0.013 &0.247 $\\pm$ 0.013 &0.137 $\\pm$ 0.009 &0.006 $\\pm$ 0.002 &0.817 $\\pm$ 0.024 &0.515 $\\pm$ 0.033 &0.323 $\\pm$ 0.021 &0.543 $\\pm$ 0.023 &0.174 $\\pm$ 0.004 &0.568 $\\pm$ 0.026 &7.176 $\\pm$ 0.212 &10.551 $\\pm$ 0.216 &479.45 $\\pm$ 19.77 &2065.8 $\\pm$ 288.843 &5.785 $\\pm$ 0.431\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & CiDer & 50 & T5 & 0.515 $\\pm$ 0.006 &0.143 $\\pm$ 0.008 &0.387 $\\pm$ 0.006 &0.308 $\\pm$ 0.006 &0.19 $\\pm$ 0.001 &0.019 $\\pm$ 0.001 &0.865 $\\pm$ 0.015 &0.726 $\\pm$ 0.018 &0.282 $\\pm$ 0.009 &0.55 $\\pm$ 0.02 &0.179 $\\pm$ 0.005 &0.576 $\\pm$ 0.014 &7.286 $\\pm$ 0.125 &10.812 $\\pm$ 0.089 &661.46 $\\pm$ 21.776 &2726.32 $\\pm$ 71.253 &7.13 $\\pm$ 0.223 \n       \\\\\n       \\cmidrule(lr){3-22}\n       & & SPider & 50 & T5 &0.393 $\\pm$ 0.008 &0.086 $\\pm$ 0.012 &0.297 $\\pm$ 0.007 &0.297 $\\pm$ 0.007 &0.183 $\\pm$ 0.007 &0.02 $\\pm$ 0.003 &0.842 $\\pm$ 0.019 &0.717 $\\pm$ 0.026 &0.297 $\\pm$ 0.019 &0.525 $\\pm$ 0.024 &0.167 $\\pm$ 0.009 &0.537 $\\pm$ 0.025 &6.986 $\\pm$ 0.262 &10.451 $\\pm$ 0.171 &530.14 $\\pm$ 16.805 &2263.4 $\\pm$ 166.221 &6.687 $\\pm$ 0.372 \n       \\\\\n  \n       \\cmidrule(lr){2-22}\n  \n       \\cmidrule(lr){2-22} \n       & Supervised &  &  & T5 & 0.503 $\\pm$ 0.001 & 0.175 $\\pm$ 0.001 & 0.411 $\\pm$ 0.001 & 0.411 $\\pm$ 0.001 & 0.309 $\\pm$ 0.001 & 0.069 $\\pm$ 0.001 & 0.929 $\\pm$ 0.000 & 1.381 $\\pm$ 0.011 & 0.443 $\\pm$ 0.001 & 0.509 $\\pm$ 0.001 & 0.101 $\\pm$ 0.001 & 0.339 $\\pm$ 0.001 & 6.531 $\\pm$ 0.006 & 10.079 $\\pm$ 0.016 & 503.600 $\\pm$ 6.530 & 2158.8 $\\pm$ 24.514 & 10.934 $\\pm$ 0.020\n       \\\\\n      \\cmidrule(lr){2-22}\n       \n       \\cmidrule(lr){2-22}\n       & Supervised + PPO & Rouge-1 & 50 & T5 & 0.537 $\\pm$ 0.004 & 0.198 $\\pm$ 0.005 & 0.433 $\\pm$ 0.002 & 0.433 $\\pm$ 0.002 & 0.314 $\\pm$ 0.003 &  0.070 $\\pm$ 0.002 & 0.930 $\\pm$ 0.001 & 1.426 $\\pm$ 0.018 & 0.449 $\\pm$ 0.001 & 0.527 $\\pm$ 0.007 & 0.112 $\\pm$ 0.001 & 0.393 $\\pm$ 0.004 &  6.680 $\\pm$ 0.044 & 10.289 $\\pm$ 0.040 & 498.2 $\\pm$ 8.931 & 2317.0 $\\pm$ 22.609 & 9.667 $\\pm$ 0.105\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & Rouge-Avg & 50 & T5 & 0.536 $\\pm$ 0.001 & 0.198 $\\pm$ 0.002 & 0.433 $\\pm$ 0.002 & 0.433 $\\pm$ 0.002 & 0.311 $\\pm$ 0.002 & 0.070 $\\pm$ 0.002 & 0.929 $\\pm$ 0.001 & 1.421 $\\pm$ 0.028 & 0.446 $\\pm$ 0.004 & 0.526 $\\pm$ 0.004 & 0.114 $\\pm$ 0.002 & 0.395 $\\pm$ 0.005 & 6.682 $\\pm$ 0.0297 & 10.274 $\\pm$ 0.042 & 506.4 $\\pm$ 6.829 & 2326.4 $\\pm$ 41.778 & 9.614 $\\pm$ 0.102\n       \\\\\n       \\cmidrule(lr){3-22}\n        \\rowcolor{lightgray}\n       & & Meteor & 50 & T5 & 0.540 $\\pm$ 0.005 & 0.204 $\\pm$ 0.005 & 0.436 $\\pm$ 0.004 & 0.436 $\\pm$ 0.004 & 0.329 $\\pm$ 0.003 & \n      0.076 $\\pm$ 0.003 & 0.930 $\\pm$ 0.001 & 1.474 $\\pm$ 0.022 & 0.447 $\\pm$ 0.004 & 0.514 $\\pm$ 0.004 & 0.105 $\\pm$ 0.002 & 0.378 $\\pm$ 0.008 & 6.631 $\\pm$ 0.053 & 10.270 $\\pm$ 0.064 & 507.0 $\\pm$ 17.146 & 2424.6 $\\pm$ 72.550 & 10.551 $\\pm$ 0.271\n       \\\\\n       \n       \\cmidrule(lr){3-22}\n       & & SPice & 50 & T5 & 0.532 $\\pm$ 0.006 & 0.194 $\\pm$ 0.007 & 0.430 $\\pm$ 0.005 & 0.430 $\\pm$ 0.005 & 0.311 $\\pm$ 0.004 & 0.068 $\\pm$ 0.003 & 0.929 $\\pm$ 0.001 & 1.415 $\\pm$ 0.029 & 0.458 $\\pm$ 0.001 & 0.532 $\\pm$ 0.008 & 0.113 $\\pm$ 0.0038 & 0.392 $\\pm$ 0.009 & 6.736 $\\pm$ 0.058 & 10.338 $\\pm$ 0.057 & 507.4 $\\pm$ 14.319 & 2313.8 $\\pm$ 27.694 & 9.742 $\\pm$ 0.208\n       \\\\\n       \n       \\cmidrule(lr){3-22}\n       & & CiDer & 50 & T5 & 0.530 $\\pm$ 0.004 & 0.191 $\\pm$ 0.003 & 0.427 $\\pm$ 0.004 & 0.427 $\\pm$ 0.004 & 0.309 $\\pm$ 0.008 & 0.063 $\\pm$ 0.002 & 0.928 $\\pm$ 0.001 & 1.337 $\\pm$ 0.040 & 0.444 $\\pm$ 0.002 & 0.518 $\\pm$ 0.009 & 0.110 $\\pm$ 0.003 & 0.382 $\\pm$ 0.006 & 6.614 $\\pm$ 0.082 & 10.166 $\\pm$ 0.053 & 490.4 $\\pm$ 9.457 & 2295.4 $\\pm$ 51.554 & 9.838 $\\pm$ 0.265 \\\\\n       \\cmidrule(lr){3-22}\n        & & SpiDer & 50 & T5 & 0.536 $\\pm$ 0.002 & 0.197 $\\pm$ 0.002 & 0.430 $\\pm$ 0.002 & 0.430 $\\pm$ 0.002 & 0.313 $\\pm$ 0.002 & 0.064 $\\pm$ 0.002 & 0.928 $\\pm$ 0.001 & 1.374 $\\pm$ 0.018 & 0.445 $\\pm$ 0.003 & 0.524 $\\pm$ 0.007 & 0.112 $\\pm$ 0.001 & 0.394 $\\pm$ 0.004 & 6.673 $\\pm$ 0.066 & 10.247 $\\pm$ 0.066 & 504.8 $\\pm$ 7.440 & 2361.8 $\\pm$ 20.856 & 9.761 $\\pm$ 0.121\n        \\\\\n      \n       \\cmidrule(lr){2-22}\n       & Supervised + NLPO & Rouge-1 & 50 & T5 & 0.545 $\\pm$ 0.002 &0.197 $\\pm$ 0.002 &0.432 $\\pm$ 0.001 &0.432 $\\pm$ 0.001 &0.31 $\\pm$ 0.002 &0.068 $\\pm$ 0.001 &0.929 $\\pm$ 0.0 &1.41 $\\pm$ 0.012 &0.449 $\\pm$ 0.001 &0.529 $\\pm$ 0.002 &0.114 $\\pm$ 0.002 &0.399 $\\pm$ 0.005 &6.705 $\\pm$ 0.018 &10.301 $\\pm$ 0.03 &498.86 $\\pm$ 8.594 &2311.46 $\\pm$ 33.451 &9.463 $\\pm$ 0.111\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & Rouge-Avg  & 50 & T5 & 0.541 $\\pm$ 0.003 &0.2 $\\pm$ 0.003 &0.435 $\\pm$ 0.002 &0.435 $\\pm$ 0.002 &0.313 $\\pm$ 0.002 &0.07 $\\pm$ 0.002 &0.93 $\\pm$ 0.001 &1.424 $\\pm$ 0.023 &0.447 $\\pm$ 0.003 &0.53 $\\pm$ 0.006 &0.113 $\\pm$ 0.002 &0.396 $\\pm$ 0.008 &6.708 $\\pm$ 0.05 &10.318 $\\pm$ 0.074 &493.64 $\\pm$ 10.068 &2319.42 $\\pm$ 55.738 &9.596 $\\pm$ 0.123 \n       \\\\\n       \\cmidrule(lr){3-22}\n       & & Meteor & 50 & T5 & 0.537 $\\pm$ 0.003 &0.201 $\\pm$ 0.004 &0.431 $\\pm$ 0.002 &0.431 $\\pm$ 0.002 &0.326 $\\pm$ 0.002 &0.074 $\\pm$ 0.003 &0.93 $\\pm$ 0.0 &1.464 $\\pm$ 0.025 &0.448 $\\pm$ 0.002 &0.516 $\\pm$ 0.006 &0.106 $\\pm$ 0.002 &0.377 $\\pm$ 0.008 &6.634 $\\pm$ 0.044 &10.26 $\\pm$ 0.077 &506.04 $\\pm$ 3.502 &2401.32 $\\pm$ 38.569 &10.453 $\\pm$ 0.194\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & SPice & 50 & T5 & 0.535 $\\pm$ 0.007 &0.193 $\\pm$ 0.008 &0.429 $\\pm$ 0.005 &0.429 $\\pm$ 0.005 &0.3 $\\pm$ 0.003 &0.064 $\\pm$ 0.002 &0.927 $\\pm$ 0.001 &1.333 $\\pm$ 0.017 &0.459 $\\pm$ 0.003 &0.553 $\\pm$ 0.013 &0.12 $\\pm$ 0.004 &0.415 $\\pm$ 0.014 &6.908 $\\pm$ 0.118 &10.445 $\\pm$ 0.057 &508.075 $\\pm$ 4.669 &2343.3 $\\pm$ 53.274 &9.249 $\\pm$ 0.225\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & CiDer & 50 & T5 & 0.533 $\\pm$ 0.003 &0.197 $\\pm$ 0.004 &0.43 $\\pm$ 0.003 &0.43 $\\pm$ 0.004 &0.316 $\\pm$ 0.004 &0.066 $\\pm$ 0.001 &0.929 $\\pm$ 0.001 &1.381 $\\pm$ 0.014 &0.446 $\\pm$ 0.004 &0.516 $\\pm$ 0.009 &0.108 $\\pm$ 0.003 &0.379 $\\pm$ 0.01 &6.583 $\\pm$ 0.077 &10.165 $\\pm$ 0.084 &490.78 $\\pm$ 9.734 &2304.52 $\\pm$ 62.068 &9.923 $\\pm$ 0.213 \n       \\\\\n       \\cmidrule(lr){3-22}\n       & & SPider & 50 & T5 & 0.532 $\\pm$ 0.006 &0.196 $\\pm$ 0.006 &0.431 $\\pm$ 0.004 &0.431 $\\pm$ 0.004 &0.314 $\\pm$ 0.004 &0.066 $\\pm$ 0.002 &0.929 $\\pm$ 0.0 &1.371 $\\pm$ 0.011 &0.448 $\\pm$ 0.002 &0.521 $\\pm$ 0.005 &0.109 $\\pm$ 0.002 &0.385 $\\pm$ 0.005 &6.623 $\\pm$ 0.034 &10.223 $\\pm$ 0.049 &485.325 $\\pm$ 5.683 &2297.575 $\\pm$ 21.271 &9.798 $\\pm$ 0.179\n       \\\\\n       \n  \n  \n     \\bottomrule\n     \\end{tabular}\n     }\n     \\caption{\\textbf{CommonGen dev evaluation}: Table shows lexical, semantic and diversity metrics for best performing models found in each algorithm-reward function combinations along with best performing supervised baseline models. Generated text from these models are submitted to official CommonGen test evaluation to obtain test scores presented in Table~\\ref{tbl:common_gen_test_scores}}\n     \\label{tbl:common_gen_dev_scores}\n  \\end{table*}\n  \\end{landscape}\n\n\n\\begin{table}[]\n\\centering\n\\footnotesize\n\\begin{tabular}{|l|r|rrr|rrr|}\n\\multicolumn{1}{|c|}{\\multirow{2}{*}{\\textbf{Algorithm}}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{Unique N}}} & \\multicolumn{3}{c|}{\\textbf{Coherence}}                                                                        & \\multicolumn{3}{c|}{\\textbf{Commonsense}}                                                                      \\\\\n\\multicolumn{1}{|c|}{}                                    & \\multicolumn{1}{c|}{}                                   & \\multicolumn{1}{c|}{\\textbf{Value}} & \\multicolumn{1}{c|}{\\textbf{Alpha}} & \\multicolumn{1}{c|}{\\textbf{Skew}} & \\multicolumn{1}{c|}{\\textbf{Value}} & \\multicolumn{1}{c|}{\\textbf{Alpha}} & \\multicolumn{1}{c|}{\\textbf{Skew}} \\\\ \\hline\nPPO+Supervised                                            & 25                                                      & \\multicolumn{1}{r|}{4.14}           & \\multicolumn{1}{r|}{0.073}          & 4.137                              & \\multicolumn{1}{r|}{4.03}           & \\multicolumn{1}{r|}{0.137}          & 4.023                              \\\\\nNLPO+Supervised                                           & 26                                                      & \\multicolumn{1}{r|}{4.25}           & \\multicolumn{1}{r|}{0.036}          & 4.253                              & \\multicolumn{1}{r|}{4.16}           & \\multicolumn{1}{r|}{0.002}          & 4.163                              \\\\\nZero Shot                                                 & 24                                                      & \\multicolumn{1}{r|}{2.15}           & \\multicolumn{1}{r|}{0.391}          & 2.154                              & \\multicolumn{1}{r|}{2.29}           & \\multicolumn{1}{r|}{0.342}          & 2.291                              \\\\\nPPO                                                       & 24                                                      & \\multicolumn{1}{r|}{2.84}           & \\multicolumn{1}{r|}{0.16}           & 2.849                              & \\multicolumn{1}{r|}{3.03}           & \\multicolumn{1}{r|}{0.081}          & 3.027                              \\\\\nSupervised                                                & 23                                                      & \\multicolumn{1}{r|}{4.39}           & \\multicolumn{1}{r|}{0.159}          & 4.387                              & \\multicolumn{1}{r|}{4.21}           & \\multicolumn{1}{r|}{0.225}          & 4.209                              \\\\\nNLPO                                                      & 24                                                      & \\multicolumn{1}{r|}{2}              & \\multicolumn{1}{r|}{0.335}          & 2.003                              & \\multicolumn{1}{r|}{2.13}           & \\multicolumn{1}{r|}{0.265}          & 2.124                             \n\\end{tabular}\n\\caption{Results of the human subject study showing the number of participants N, average Likert scale value for coherence and sentiment, Krippendorf's alpha showing inter-annotator agreement, and Skew. For each model a total of 100 samples were drawn randomly from the test set and rated by 3 annotators each, resulting in 300 data points per algorithm.}\n\\label{app:commongen:human_agreement}\n\\end{table}\n\n\n\n\\begin{table}[]\n\\footnotesize\n\\centering\n\\begin{tabular}{|l|l|rr|rr|}\n\\multicolumn{1}{|c|}{\\multirow{2}{*}{\\textbf{Group 1}}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{Group 2}}} & \\multicolumn{2}{c|}{\\textbf{Coherence}}                                                      & \\multicolumn{2}{c|}{\\textbf{Commonsense}}                                                    \\\\\n\\multicolumn{1}{|c|}{}                                  & \\multicolumn{1}{c|}{}                                  & \\multicolumn{1}{c|}{\\textbf{Diff (G2-G1)}} & \\multicolumn{1}{c|}{\\textit{\\textbf{p-values}}} & \\multicolumn{1}{c|}{\\textbf{Diff (G2-G1)}} & \\multicolumn{1}{c|}{\\textit{\\textbf{p-values}}} \\\\ \\hline\nNLPO                                                    & PPO                                                    & \\multicolumn{1}{r|}{\\textbf{0.847}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{0.897}}        & \\textbf{0.001}                                  \\\\\nNLPO                                                    & Supervised                                             & \\multicolumn{1}{r|}{\\textbf{2.397}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{2.083}}        & \\textbf{0.001}                                  \\\\\nNLPO                                                    & NLPO+Supervised                                        & \\multicolumn{1}{r|}{\\textbf{2.257}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{2.033}}        & \\textbf{0.001}                                  \\\\\nNLPO                                                    & PPO+Supervised                                         & \\multicolumn{1}{r|}{\\textbf{2.143}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.897}}        & \\textbf{0.001}                                  \\\\\nNLPO                                                    & Zero Shot                                              & \\multicolumn{1}{r|}{0.153}                 & 0.515                                           & \\multicolumn{1}{r|}{0.157}                 & 0.624                                           \\\\\nPPO                                                     & Supervised                                             & \\multicolumn{1}{r|}{\\textbf{1.550}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.187}}        & \\textbf{0.001}                                  \\\\\nPPO                                                     & NLPO+Supervised                                        & \\multicolumn{1}{r|}{\\textbf{1.410}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.137}}        & \\textbf{0.001}                                  \\\\\nPPO                                                     & PPO+Supervised                                         & \\multicolumn{1}{r|}{\\textbf{1.297}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.000}}        & \\textbf{0.001}                                  \\\\\nPPO                                                     & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-0.693}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.740}}       & \\textbf{0.001}                                  \\\\\nSupervised                                              & NLPO+Supervised                                        & \\multicolumn{1}{r|}{-0.140}                & 0.601                                           & \\multicolumn{1}{r|}{-0.050}                & 0.900                                           \\\\\nSupervised                                              & PPO+Supervised                                         & \\multicolumn{1}{r|}{\\textbf{-0.253}}       & \\textbf{0.050}                                  & \\multicolumn{1}{r|}{\\textbf{-0.187}}       & \\textbf{0.045}                                  \\\\\nSupervised                                              & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-2.243}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-1.927}}       & \\textbf{0.001}                                  \\\\\nNLPO+Supervised                                         & PPO+Supervised                                         & \\multicolumn{1}{r|}{\\textbf{-0.113}}       & \\textbf{0.008}                                  & \\multicolumn{1}{r|}{\\textbf{-0.137}}       & \\textbf{0.007}                                  \\\\\nNLPO+Supervised                                         & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-2.103}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-1.877}}       & \\textbf{0.001}                                  \\\\\nPPO+Supervised                                          & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-1.990}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-1.740}}       & \\textbf{0.001}                                 \n\\end{tabular}\n\\caption{Results of an post-hoc Tukey HSD Test for difference in means between pairs of algorithms (Group 2 - Group 1) and corresponding $p$-values. Individually statistically significant results are bolded and are used to discuss results in the analysis. Overall $p$-values showing that there is a significant difference in means between the models via a one-way ANOVA test are significant with $p \\ll 0.05$ for both coherence and sentiment.}\n\\label{app:commongen:human_tukey}\n\\end{table}\n\n\n\n\\clearpage\n\n\\subsubsection{Human Participant Study}\n\nFigure~\\ref{fig:description_interface_commongen} shows the commongen instructions, examples, and interface used for the human evaluation experiments. Different from the other human evaluations, we didn't provide any prompt because knowing the set of words to be used isn't required for rating either of the axes.\nTables~\\ref{app:commongen:human_agreement},~\\ref{app:commongen:human_tukey} show averaged results, annotator agreement, and the results of statistical significance tests to determine which models output better generations when rated by humans.\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=.45\\linewidth]{figures\/commongen_human_study\/commongen_human1.png}\n    \\includegraphics[width=.45\\linewidth]{figures\/commongen_human_study\/commongen_human2.png}\n    \\includegraphics[width=.45\\linewidth]{figures\/commongen_human_study\/commongen_human3.png}\n    \\caption{Instructions, examples, and interface for the Commongen task.}\n    \\label{fig:description_interface_commongen}\n\\end{figure*}\n\n\n\\subsubsection{Qualitative Analysis}\n\\label{app:tbl_common_gen_qualitative}\nThis section shows sample generations from different algorithms for three randomly picked prompts.\n\n\\begin{lstlisting}\nSample 1\nPrompt: generate a sentence with: apron cut hat kitchen sausage\nZero-Shot: generate a sentence with: apron cut hat kitchen sausage.\nReferences:\nPPO: sausage in the kitchen on an apron.\nNLPO: sausage cut hat cut hat cut hat cut apron cut hat\nSupervised: A man is cutting sausage on an apron in a kitchen.\nSupervised + PPO: A man cutting sausage from a hat on an apron in the kitchen.\nSupervised + NLPO: A man cuts sausage in a hat on an apron in the kitchen.\n\nSample 2\nPrompt: generate a sentence with: backhand competition player reach shuttlecock\nZero-Shot: backhand competition player reach shuttlecock.\nPPO: the ball in the backhand and the competition player in the shuttlecock.\nNLPO: backhanded competition player reaches shuttlecock.\nSupervised: tennis player reaches for a shuttlecock during a competition.\nSupervised + PPO: football player reaches for a shuttlecock after hitting a backhand during the competition.\nSupervised + NLPO: football player reaches for a shuttlecock in backhand during the competition.\n\nSample 3\nPrompt: generate a sentence with: bowl combine cucumber onion pasta\nZero-Shot: cucumber onion pasta.\nPPO: the cucumber onion pasta in a small bowl.\nNLPO: the in a bowl Combine cucumber onion pasta in a large bowl. Toss pasta\nSupervised: A bowl containing pasta, tomatoes, cucumbers, and onions.\nSupervised + PPO: A bowl containing pasta topped with cucumbers, onions, and peppers.\nSupervised + NLPO: A bowl containing a mixture of pasta, cucumber, and onion.\n\\end{lstlisting}\n\n\\subsubsection{Setup} As a representative of the summarization task, we consider CNN\/DM dataset consisting of long news articles and their highlights written by news authors. The dataset consists of 287k training, 13k validation and 11k test examples. We trained RL methods using 3 different automated metrics, namely Rouge-1, Rouge-avg and Meteor. We chose T5 as our base LM as it is pre-trained in a unified text-to-text framework and relishes Zero-Shot capabilities. For decoding, we use multinomial sampling with a temperature of $0.7$ for all the models.\n\n\\begin{table*}[ht!]\n\\centering\n\\footnotesize\n\n\\resizebox{0.5\\textwidth}{!}{\n\\begin{tabular}{ll}\n\\toprule\n\\textbf{Model Params}\n& \\multicolumn{1}{c}{\\textbf{value}} \\\\ \n\\cmidrule{1-2}\nsupervised & batch size: $16$\\\\\n& epochs: $2$ \\\\\n& learning rate: $0.0001$ \\\\\n& learning rate scheduler: cosine \\\\\n& weight decay: $0.1$ \\\\\n\\cmidrule{1-2}\nppo\/ \\textcolor{forestgreen2}{nlpo} & steps per update: $5120$\\\\\n    & total number of steps: $512000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$ \\\\\n    & learning rate: $0.000002$ \\\\\n    & entropy coefficient: $0.0$ \\\\\n    & initial kl coeff: $0.001$ \\\\\n    & target kl: $0.2$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & rollouts top k: sweep of ($50$,$100$) \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.9$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: sweep of ($10$, $20$, $30$)} \\\\\n\\cmidrule{1-2}\nsupervised+ppo\/ \\textcolor{forestgreen2}{nlpo} & steps per update: 5120\\\\\n    & total number of steps: $256000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$ \\\\\n    & learning rate: $0.000002$ \\\\\n    & entropy coefficient: $0.0$ \\\\\n    & initial kl coeff: $0.01$ \\\\\n    & target kl: $0.2$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & rollouts top k: sweep of ($50$,$100$) \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.9$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: sweep of ($10$, $20$, $30$)} \\\\\n\\cmidrule{1-2}\ndecoding & sampling: True \\\\\n& temperature: $0.7$ \\\\\n& min length: $50$ \\\\\n& max new tokens: $100$\\\\\n\\cmidrule{1-2}\ntokenizer & padding side: left\\\\\n& truncation side: right\\\\\n& max length: 512 \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{CNN\/DM Hyperparams}: Table shows a list of all hyper-parameters and their settings}\n       \\label{tbl:imdb_gen_hyperparams}\n\\end{table*}\n\n\\subsubsection{Results and Discussion}\n\nTable~\\ref{tbl:summ_lexical_scores} presents benchmarking results on test set reporting a wide range of metrics: lexical, semantic, factual correctness and diversity metrics. As baselines, we report lead-3 which selects first three sentences as the summary, Zero-Shot and a supervised model. PPO and NLPO models are on par with supervised performance on several metrics including Rouge-2, Rouge-L, and Bleu. On fine-tuning on top of supervised model, performance improves consistently on all metrics indicating that RL fine-tuning is beneficial. Another interesting finding is that, RL fine-tuned models are factually consistent as measured by SummaCZS metric. For ablations on PPO params, NLPO params, we refer to Tables \\ref{tbl:summ_ppo_ablations},\\ref{tbl:summ_nlpo_ablation}.\n\n\\begin{landscape}\n\\begin{table}\n   \\centering\n   \\resizebox{1.5\\textwidth}{!}{\n   \\begin{tabular}{@{}cccc|ccccccc|c|ccccccccc@{}}\n   \\toprule\n     Tasks \n     & \\multicolumn{3}{c}{\\textbf{\\_}} \n     & \\multicolumn{7}{c}{\\textbf{Lexical and Semantic Metrics}} \n     & \\multicolumn{1}{c}{\\textbf{Factual Consistency}} \n     & \\multicolumn{8}{c}{\\textbf{Diversity Metrics}}\n     \\\\\n     & Alg & Reward Function & LM & Rouge-1 & Rouge-2 & Rouge-L & Rouge-LSum & Meteor & BLEU & BertScore & SummaCZS & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ & Mean Output Length\n     \\\\\n     \\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-9} \\cmidrule(lr){10-10} \\cmidrule(lr){11-11} \\cmidrule(lr){12-12}\\cmidrule(lr){13-13}\\cmidrule(lr){14-14} \\cmidrule(lr){15-15} \\cmidrule(lr){16-16}  \\cmidrule(lr){17-17}  \\cmidrule(lr){18-18} \\cmidrule(lr){19-19}\\cmidrule(lr){20-20}\n     \\\\\n     \\multirow{31}{*}{CNN\/DM} & Lead-3 &  & & 0.401 & 0.175 & 0.250 & 0.363 & 0.333 & 0.099 & 0.874 & 0.993 & 0.750 & 0.0482 & 0.386 & 10.481 & 16.631 & 21465 & 273153 & 84\n     \\\\\n\n     \\cmidrule(lr){2-20}\n     & Zero-Shot &  & T5 & 0.372 & 0.145 & 0.247 & 0.311 & 0.256 & 0.077 & 0.864 & 0.654 & 0.725 & 0.061 & 0.414 & 10.285 & 16.183 & 19113 & 193999 & 55\n     \\\\\n        \n     \\cmidrule(lr){2-20} \n     \\rowcolor{lightgray}\n     & PPO & Rouge-1  & T5 & 0.410 & 0.182 & 0.283 & 0.349 & 0.276 & 0.095 & 0.876 & 0.622 & 0.760 & 0.068 & 0.464 & 10.661 & 16.437 & 18189 & 191383 & 47\n     \\\\\n    \n     \\cmidrule(lr){3-20}\n     \\\\\n     &  & Rouge-Avg & T5 &  0.396 & 0.176 & 0.273 & 0.338 & 0.270 & 0.095 & 0.874 & 0.622 & 0.773 & 0.071 & 0.490 & 10.830 & 16.664 & 19478 & 209140 & 48 \n     \\\\\n    \n     \\cmidrule(lr){3-20}\n     & & Meteor & T5 & 0.408 & 0.178 & 0.276 & 0.342 & 0.301 & 0.109 & 0.873 & 0.527 & 0.765 & 0.060 & 0.447 & 10.699 & 16.688 & 20528 & 234386 & 61\n     \\\\\n     \\cmidrule(lr){2-20} \n     & NLPO & Rouge-1 & T5 & 0.404 & 0.180 & 0.278 & 0.344 & 0.275 & 0.096 & 0.875 & 0.636 & 0.771 & 0.069 & 0.480 & 10.789 & 16.618 & 18677 & 201971 & 48\n     \\\\\n     \\cmidrule(lr){3-20}\n     & & Rouge-Avg  & T5 & 0.404 & 0.177 & 0.279 & 0.344 & 0.274 & 0.094 & 0.874 & 0.586 & 0.765 & 0.066 & 0.476 & 10.744 & 16.620 & 18179 & 206368 & 50\n     \\\\\n     \\cmidrule(lr){3-20}\n     \\rowcolor{lightgray}\n     & & Meteor & T5 & 0.405 & 0.180 & 0.277 & 0.343 & 0.292 & 0.108 & 0.872 & 0.578 & 0.772 & 0.064 & 0.471 & 10.802 & 16.766 & 20212 & 231038 & 56\\\\ \n\n     \\cmidrule(lr){2-20}\n\n     & Supervised &  & T5 & 0.411 & 0.177 & 0.276 & 0.343 & 0.309 & 0.108 & 0.876 & 0.654 & 0.727 & 0.057 & 0.401 & 10.459 & 16.410 & 21096 & 230343 & 68 \n     \\\\\n     \\cmidrule(lr){2-20} \n     \n     & Supervised + PPO & Rouge-1 & T5 & 0.417 & 0.189 & 0.294 & 0.358 & 0.278 & 0.101 & 0.882 & 0.722 & 0.750 & 0.070 & 0.459 & 10.595 & 16.389 & 18184 & 184220 & 46\n     \\\\\n     \\cmidrule(lr){3-20}\n     & & Rouge-Avg & T5 & 0.425 & 0.194 & 0.297 & 0.363 & 0.296 & 0.114 & 0.882 & 0.728 & 0.747 & 0.066 & 0.445 & 10.589 & 16.458 & 18939 & 200617 & 52 \n     \\\\\n      \\cmidrule(lr){3-20}\n      \\rowcolor{lightgray}\n     & & Meteor & T5 & 0.426 & 0.194 & 0.293 & 0.361 & 0.316 & 0.125 & 0.880 & 0.726 & 0.741 & 0.059 & 0.420 & 10.532 & 16.491 & 20395 & 224432 & 63 \n     \\\\ \n\n     \\cmidrule(lr){2-20}\n     & Supervised + NLPO & Rouge-1 & T5 & 0.421 & 0.193 & 0.297 & 0.361 & 0.287 & 0.108 & \\textbf{0.882} & 0.740 & 0.748 & 0.067 & 0.446 & 10.528 & 16.313 & 18204 & 185561 & 48\n     \\\\\n     \\cmidrule(lr){3-20}\n     & & Rouge-Avg  & T5 & 0.424 & 0.193 & 0.296 & 0.363 & 0.295 & 0.115 & 0.882 & 0.743 & 0.744 & 0.065 & 0.443 & 10.570 & 16.444 & 18747 & 201705 & 53\n     \\\\\n     \\cmidrule(lr){3-20}\n     \\rowcolor{lightgray}\n     & & Meteor & T5  & \\textbf{0.429} & 0.194 & 0.293 & 0.361 & \\textbf{0.319} & 0.124 & 0.880 & 0.743 & 0.745 & 0.059 & 0.422 & 10.574 & 16.516 & 20358 & 226801 & 63\n     \\\\\n   \\bottomrule\n   \\end{tabular}\n   }\n   \\caption{\\textbf{CNN\/Daily Mail test evaluation}: Table presents a wide range of metrics: lexical, semantic, factual correctness and diversity metrics on test set. As baselines, we report lead-3 which selects first three sentences as the summary, Zero-Shot and a supervised model. PPO and NLPO models are on par with supervised performance on several metrics including Rouge-2, Rouge-L, and Bleu. On fine-tuning on top of supervised model, performance improves consistently on all metrics indicating that RL fine-tuning is beneficial. Another interesting finding is that, RL fine-tuned models are factually consistent as measured by SummaCZS metric.}\n   \\label{tbl:summ_lexical_scores}\n\\end{table}\n\\end{landscape}\n\n\\begin{table*}[h]\n\\centering\n   \\resizebox{0.7\\textwidth}{!}{\n   \\begin{tabular}{@{}cccccccccc@{}}\n   \\toprule\n     \\multicolumn{3}{c}{\\textbf{\\_}} \n     & \\multicolumn{7}{c}{\\textbf{Lexical and Semantic Metrics}} \n     \\\\\n     Alg & Reward Function & Top k & Rouge-1 & Rouge-2 & Rouge-L & Rouge-LSum & Meteor & BLEU & BertScore \\\\\n     \\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-10}\\\\\n     \\multirow{6}{*}{PPO} & Rouge-1 & 50 & 0.404 & 0.181 & 0.280 & 0.346 & 0.273 & \\textbf{0.095} & 0.874\\\\\n     \\rowcolor{lightgray} \n     & & 100 & \\textbf{0.412} & \\textbf{0.186} & \\textbf{0.286} & \\textbf{0.354} & \\textbf{0.276} & 0.094 & \\textbf{0.876} \\\\\n     \\cmidrule(lr){2-10}\n     & Rouge-Avg & 50 & \\textbf{0.401} & 0.177 & \\textbf{0.276} & 0.342 & \\textbf{0.271} & 0.092 & 0.873 \\\\\n     \\rowcolor{lightgray} \n     & &  100 & 0.399 & \\textbf{0.179} & 0.275 & \\textbf{0.342} & 0.270 & \\textbf{0.094} & \\textbf{0.874} \\\\\n     \\cmidrule(lr){2-10}\n     \\rowcolor{lightgray} \n     & Meteor & 50 &  \\textbf{0.413} & \\textbf{0.182} & \\textbf{0.279} & \\textbf{0.348} & \\textbf{0.301} & \\textbf{0.110} & \\textbf{0.873} \\\\\n     & &  100 & 0.409 & 0.179 & 0.276 & 0.345 & 0.296 & 0.108 & 0.871 \\\\\n      \\cmidrule(lr){1-10}\n     \\multirow{6}{*}{Supervised+PPO} & Rouge-1 & 50 & 0.414 & 0.190 & 0.293 & 0.358 & 0.272 & 0.097 & 0.881 \\\\\n     \\rowcolor{lightgray} \n     & &  100 & \\textbf{0.420} & \\textbf{0.193} & \\textbf{0.295} & \\textbf{0.362} & \\textbf{0.277} & \\textbf{0.100} & \\textbf{0.881} \\\\\n     \\cmidrule(lr){2-10}\n     & Rouge-Avg & 50 & 0.426 & 0.196 & 0.298 & 0.366 & 0.294 & \\textbf{0.114} & 0.881 \\\\\n     \\rowcolor{lightgray}\n     & &  100 & \\textbf{0.427} & \\textbf{0.196} & \\textbf{0.298} & \\textbf{0.366} & \\textbf{0.294} & 0.113 & \\textbf{0.881} \\\\\n     \\cmidrule(lr){2-10}\n     & Meteor & 50 & 0.429 & 0.197 & 0.297 & 0.367 & 0.306 & 0.122 & \\textbf{0.881} \\\\\n     \\rowcolor{lightgray}\n     & &  100 & \\textbf{0.432} & \\textbf{0.199} & \\textbf{0.297} & \\textbf{0.367} & \\textbf{0.317} & \\textbf{0.131} & 0.879 \\\\\n     \\cmidrule(lr){1-10}\n    \n   \\bottomrule\n   \\end{tabular}\n   }\n   \\caption{\\textbf{PPO Ablation\/Model Selection}: Evaluation of PPO models on validation set with different reward functions and top k values for rollouts. For each alg-reward combo, best model (top k ) is chosen. }\n   \\label{tbl:summ_ppo_ablations}\n\\end{table*}\n\n\\begin{table*}[h]\n\\centering\n   \\resizebox{0.9\\textwidth}{!}{\n   \\begin{tabular}{@{}cccccccccccc@{}}\n   \\toprule\n     \\multicolumn{5}{c}{\\textbf{\\_}} \n     & \\multicolumn{7}{c}{\\textbf{Lexical and Semantic Metrics}} \n     \\\\\n     Alg & Reward Function & Top k (rollout) & Top p (Action mask) & target update $n_iters$ & Rouge-1 & Rouge-2 & Rouge-L & Rouge-LSum & Meteor & BLEU & BertScore \\\\\n     \\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4} \\cmidrule(lr){5-5}\\cmidrule(lr){6-12}\\\\\n     \\multirow{18}{*}{NLPO} & Rouge-1 & 50 & 0.9 & 10 & \n      0.400 & 0.178 & 0.275 & 0.343 & 0.269 & 0.094 & 0.872\\\\\n     & & & & 20 & 0.396 & 0.173 & 0.274 & 0.340 & 0.257 & 0.082 & 0.873 \\\\\n     & & & & 30 & 0.396 & 0.174 & 0.273 & 0.339 & 0.265 & 0.091 & 0.872 \\\\\n     & & 100 & 0.9 & 10 & \\textbf{0.407} & 0.177 & 0.279 & 0.347 & 0.265 & 0.085 & \\textbf{0.875} \\\\\n      \\rowcolor{lightgray} \n     & & & & 20 & 0.406 & \\textbf{0.182} & \\textbf{0.281} & \\textbf{0.347} & \\textbf{0.273} & \\textbf{0.094} & 0.874\\\\\n     & & & & 30 & 0.405 & 0.180 & 0.279 & 0.347 & 0.269 & 0.091 & 0.875\\\\\n     \\cmidrule(lr){2-12}\n     \n      & Rouge-Avg & 50 & 0.9 & 10 & 0.400 & \\textbf{0.180} & 0.276 & 0.343 & 0.271 & 0.096 & 0.873\\\\\n      & & & & 20 & 0.349 & 0.147 & 0.241 & 0.298 & 0.237 & 0.078 & 0.858\\\\\n      & & & & 30 & 0.393 & 0.173 & 0.272 & 0.336 & 0.267 & 0.092 & 0.870\\\\\n      & & 100 & 0.9 & 10 & 0.396 & 0.174 & 0.274 & 0.339 & 0.265 & 0.088 & 0.872\\\\\n      \\rowcolor{lightgray} \n      & & & & 20 & \\textbf{0.406} & 0.179 & \\textbf{0.280} & \\textbf{0.347} &\n      \\textbf{0.272} & \\textbf{0.092} & \\textbf{0.874}\\\\\n      & & & & 30 & 0.400 & 0.178 & 0.279 & 0.344 & 0.266 & 0.087 & 0.874\\\\\n    \n      \\cmidrule(lr){2-12}\n      & Meteor & 50 & 0.9 & 10 & 0.404 & 0.177 & 0.274 & 0.343 & 0.286 & 0.102 & 0.872\\\\\n      & & & & 20 & 0.406 & 0.180 & 0.276 & 0.343 & 0.292 & 0.107 & 0.871\\\\\n      & & & & 30 & 0.401 & 0.172 & 0.271 & 0.337 & 0.288 & 0.099 & 0.870\\\\\n      & & 100 & 0.9 & 10 & 0.405 & 0.178 & 0.276 & 0.343 & \\textbf{0.294} & 0.107 & 0.870\\\\\n      & & & & 20 & 0.406 & 0.176 & 0.276 & 0.343 & 0.291 & 0.106 & 0.872\\\\\n      \\rowcolor{lightgray} \n      & & & & 30 & \\textbf{0.409} & \\textbf{0.184} & \\textbf{0.280} & \\textbf{0.348} & 0.291 & \\textbf{0.108} & \\textbf{0.873}\\\\\n      \n      \\cmidrule(lr){1-12}\n      \n     \n     \\multirow{18}{*}{Supervised + NLPO} & Rouge-1 & 50 & 0.9 & 10 & \\textbf{0.425} & 0.196 & \\textbf{0.299} & \\textbf{0.366} & 0.285 & 0.106 & \\textbf{0.882} \\\\\n     & & &  & 20 & 0.417 & 0.191 & 0.295 & 0.360 & 0.276 & 0.100 & 0.881 \\\\\n     & & & & 30 & 0.418 & 0.192 & 0.296 & 0.361 & 0.278 & 0.101 & 0.881 \\\\\n     & & 100 & 0.9 & 10 &  0.424 & 0.196 & 0.299 & 0.366 & 0.286 & 0.106 & 0.882 \\\\\n     & & & & 20 & 0.423 & 0.196 & 0.299 & 0.365 & \\textbf{0.289} & \\textbf{0.110} & 0.881 \\\\\n     & & & & 30 & 0.420 & 0.193 & 0.296 & 0.362 & 0.279 & 0.102 & 0.881 \\\\\n    \\cmidrule(lr){2-12}\n     & Rouge-Avg & 50 & 0.9 & 10 & 0.426 & 0.197 & 0.298 & 0.367 & 0.294 & 0.115 & 0.881 \\\\\n     & & & & 20 &  0.425 & 0.196 & 0.298 & 0.366 & 0.292 & 0.112 & 0.881\\\\\n     & & & & 30 & 0.424 & 0.194 & 0.297 & 0.365 & 0.287 & 0.107 & 0.881 \\\\\n     & & 100 & 0.9 & 10 & 0.424 & 0.196 & 0.298 & 0.365 & 0.291 & 0.113 & 0.881\\\\\n     & & & & 20 & 0.428 & 0.198 & 0.300 & 0.368 & 0.296 & 0.115 & 0.882\\\\\n     \\rowcolor{lightgray} \n     & & & & 30 & \\textbf{0.429} & \\textbf{0.199} & \\textbf{0.300} & \\textbf{0.369} & \\textbf{0.296} & \\textbf{0.116} & \\textbf{0.882}\\\\\n     \\cmidrule(lr){2-12}\n     & Meteor & 50 & 0.9 & 10 &  0.430 & 0.197 & 0.294 & 0.364 & 0.320 & 0.130 & 0.879\\\\\n     & & & & 20 & 0.432 & 0.198 & 0.297 & 0.367 & 0.318 & 0.130 & 0.880\\\\\n     & & & & 30 & 0.423 & 0.191 & 0.293 & 0.361 & 0.297 & 0.116 & 0.879 \\\\\n     \\rowcolor{lightgray} \n     & & 100 & 0.9 & 10 & \\textbf{0.435} & \\textbf{0.200} & \\textbf{0.298} & \\textbf{0.369} & \\textbf{0.320} & 0.131 & \\textbf{0.881} \\\\\n     & & & & 20 &  0.433 & 0.198 & 0.297 & 0.368 & 0.319 & 0.130 & 0.879 \\\\\n     & & & & 30 & 0.434 & 0.200 & 0.297 & 0.369 & 0.324 & \\textbf{0.132} & 0.879 \\\\\n   \\bottomrule\n   \\end{tabular}\n   }\n   \\caption{\\textbf{NLPO Ablation\/Model Selection}: Evaluation of NLPO models on validation set with different reward functions, top k values for rollouts and target update iterations. For each alg-reward combo, best model is chosen}\n   \\label{tbl:summ_nlpo_ablation}\n\\end{table*}\n\n\n \n\n\\begin{table}[]\n\\centering\n\\footnotesize\n\\begin{tabular}{|l|r|rrr|rrr|}\n\\multicolumn{1}{|c|}{\\multirow{2}{*}{\\textbf{Algorithm}}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{Unique N}}} & \\multicolumn{3}{c|}{\\textbf{Coherence}}                                                                        & \\multicolumn{3}{c|}{\\textbf{Quality}}                                                                          \\\\\n\\multicolumn{1}{|c|}{}                                    & \\multicolumn{1}{c|}{}                                   & \\multicolumn{1}{c|}{\\textbf{Value}} & \\multicolumn{1}{c|}{\\textbf{Alpha}} & \\multicolumn{1}{c|}{\\textbf{Skew}} & \\multicolumn{1}{c|}{\\textbf{Value}} & \\multicolumn{1}{c|}{\\textbf{Alpha}} & \\multicolumn{1}{c|}{\\textbf{Skew}} \\\\ \\hline\nPPO+Supervised                                            & 22                                                      & \\multicolumn{1}{r|}{4.21}           & \\multicolumn{1}{r|}{0.198}          & 4.224                              & \\multicolumn{1}{r|}{3.97}           & \\multicolumn{1}{r|}{0.256}          & 3.98                               \\\\\nNLPO+Supervised                                           & 19                                                      & \\multicolumn{1}{r|}{4.3}            & \\multicolumn{1}{r|}{0.26}           & 4.308                              & \\multicolumn{1}{r|}{3.98}           & \\multicolumn{1}{r|}{0.089}          & 4                                  \\\\\nZero Shot                                                 & 17                                                      & \\multicolumn{1}{r|}{3.73}           & \\multicolumn{1}{r|}{0.1}            & 3.757                              & \\multicolumn{1}{r|}{3.69}           & \\multicolumn{1}{r|}{0.25}           & 3.722                              \\\\\nSupervised                                                & 19                                                      & \\multicolumn{1}{r|}{4.25}           & \\multicolumn{1}{r|}{0.116}          & 4.241                              & \\multicolumn{1}{r|}{3.99}           & \\multicolumn{1}{r|}{0.2}            & 3.986                              \\\\\nNLPO                                                      & 17                                                      & \\multicolumn{1}{r|}{4.03}           & \\multicolumn{1}{r|}{0.13}           & 4.042                              & \\multicolumn{1}{r|}{3.83}           & \\multicolumn{1}{r|}{0.191}          & 3.832                              \\\\\nPPO                                                       & 21                                                      & \\multicolumn{1}{r|}{3.94}           & \\multicolumn{1}{r|}{0.111}          & 3.945                              & \\multicolumn{1}{r|}{3.76}           & \\multicolumn{1}{r|}{0.129}          & 3.767                              \\\\\nHuman                                                     & 19                                                      & \\multicolumn{1}{r|}{3.89}           & \\multicolumn{1}{r|}{0.277}          & 3.902                              & \\multicolumn{1}{r|}{3.77}           & \\multicolumn{1}{r|}{0.029}          & 3.769                             \n\\end{tabular}\n\\caption{Results of the human subject study showing the number of participants N, average Likert scale value for coherence and sentiment, Krippendorf's alpha showing inter-annotator agreement, and Skew. For each model a total of 50 samples were drawn randomly from the test set and rated by 3 annotators each, each resulting in 150 data points per algorithm.}\n\\label{app:summariazation:human_agreement}\n\\end{table}\n\n\n\\begin{table}[]\n\\centering\n\\footnotesize\n\\begin{tabular}{|l|l|rr|rr|}\n                                       &                                       & \\multicolumn{2}{c|}{\\textbf{Coherence}}                                                      & \\multicolumn{2}{c|}{\\textbf{Quality}}                                                        \\\\\n\\multicolumn{1}{|c|}{\\textbf{Group 1}} & \\multicolumn{1}{c|}{\\textbf{Group 2}} & \\multicolumn{1}{c|}{\\textbf{Diff (G2-G1)}} & \\multicolumn{1}{c|}{\\textit{\\textbf{p-values}}} & \\multicolumn{1}{c|}{\\textbf{Diff (G2-G1)}} & \\multicolumn{1}{c|}{\\textit{\\textbf{p-values}}} \\\\ \\hline\nHuman                                  & NLPO                                  & \\multicolumn{1}{r|}{0.147}                 & 0.755                                           & \\multicolumn{1}{r|}{0.060}                 & 0.900                                           \\\\\nHuman                                  & NLPO+Supervised                       & \\multicolumn{1}{r|}{\\textbf{0.413}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{0.213}}        & \\textbf{0.047}                                  \\\\\nHuman                                  & PPO                                   & \\multicolumn{1}{r|}{0.053}                 & 0.900                                           & \\multicolumn{1}{r|}{-0.007}                & 0.900                                           \\\\\nHuman                                  & PPO+Supervised                        & \\multicolumn{1}{r|}{\\textbf{0.327}}        & \\textbf{0.024}                                  & \\multicolumn{1}{r|}{0.200}                 & 0.544                                           \\\\\nHuman                                  & Supervised                            & \\multicolumn{1}{r|}{\\textbf{0.360}}        & \\textbf{0.008}                                  & \\multicolumn{1}{r|}{\\textbf{0.220}}        & \\textbf{0.043}                                  \\\\\nHuman                                  & Zero Shot                             & \\multicolumn{1}{r|}{-0.160}                & 0.679                                           & \\multicolumn{1}{r|}{-0.080}                & 0.900                                           \\\\\nNLPO                                   & NLPO+Supervised                       & \\multicolumn{1}{r|}{\\textbf{0.267}}        & \\textbf{0.012}                                  & \\multicolumn{1}{r|}{\\textbf{0.153}}        & \\textbf{0.008}                                  \\\\\nNLPO                                   & PPO                                   & \\multicolumn{1}{r|}{-0.093}                & 0.900                                           & \\multicolumn{1}{r|}{-0.067}                & 0.900                                           \\\\\nNLPO                                   & PPO+Supervised                        & \\multicolumn{1}{r|}{0.180}                 & 0.564                                           & \\multicolumn{1}{r|}{0.140}                 & 0.860                                           \\\\\nNLPO                                   & Supervised                            & \\multicolumn{1}{r|}{0.213}                 & 0.361                                           & \\multicolumn{1}{r|}{0.160}                 & 0.754                                           \\\\\nNLPO                                   & Zero Shot                             & \\multicolumn{1}{r|}{\\textbf{-0.307}}       & \\textbf{0.044}                                  & \\multicolumn{1}{r|}{-0.140}                & 0.860                                           \\\\\nNLPO+Supervised                        & PPO                                   & \\multicolumn{1}{r|}{\\textbf{-0.360}}       & \\textbf{0.008}                                  & \\multicolumn{1}{r|}{\\textbf{-0.220}}       & \\textbf{0.043}                                  \\\\\nNLPO+Supervised                        & PPO+Supervised                        & \\multicolumn{1}{r|}{\\textbf{-0.087}}       & \\textbf{0.009}                                  & \\multicolumn{1}{r|}{\\textbf{-0.013}}       & \\textbf{0.009}                                  \\\\\nNLPO+Supervised                        & Supervised                            & \\multicolumn{1}{r|}{\\textbf{-0.053}}       & \\textbf{0.009}                                  & \\multicolumn{1}{r|}{0.007}                 & 0.900                                           \\\\\nNLPO+Supervised                        & Zero Shot                             & \\multicolumn{1}{r|}{\\textbf{-0.573}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.293}}       & \\textbf{0.012}                                  \\\\\nPPO                                    & PPO+Supervised                        & \\multicolumn{1}{r|}{0.273}                 & 0.106                                           & \\multicolumn{1}{r|}{0.207}                 & 0.508                                           \\\\\nPPO                                    & Supervised                            & \\multicolumn{1}{r|}{0.307}                 & 0.044                                           & \\multicolumn{1}{r|}{0.227}                 & 0.394                                           \\\\\nPPO                                    & Zero Shot                             & \\multicolumn{1}{r|}{-0.213}                & 0.361                                           & \\multicolumn{1}{r|}{-0.073}                & 0.900                                           \\\\\nPPO+Supervised                         & Supervised                            & \\multicolumn{1}{r|}{0.033}                 & 0.900                                           & \\multicolumn{1}{r|}{0.020}                 & 0.900                                           \\\\\nPPO+Supervised                         & Zero Shot                             & \\multicolumn{1}{r|}{-0.487}                & 0.001                                           & \\multicolumn{1}{r|}{-0.280}                & 0.155                                           \\\\\nSupervised                             & Zero Shot                             & \\multicolumn{1}{r|}{-0.520}                & 0.001                                           & \\multicolumn{1}{r|}{-0.300}                & 0.101                                          \n\\end{tabular}\n\\caption{Results of an post-hoc Tukey HSD Test for difference in means between pairs of algorithms (Group 2 - Group 1) and corresponding $p$-values. Individually statistically significant results are bolded and are used to discuss results in the analysis. Overall $p$-values showing that there is a significant difference in means between the models via a one-way ANOVA test are significant with $p \\ll 0.05$ for both coherence and sentiment.}\n\\label{app:summariazation:human_tukey}\n\\end{table}\n\n\n\\clearpage\n\n\\subsubsection{Human Participant Study}\n\nFigure~\\ref{fig:description_interface_summarization} shows the summarization instructions and interface used for the human evaluation experiments. Participants weren't required to read the entire article, but to encourage some reading, a minimum time on the window of 15s was enforced via hiding the sliders.\nTables~\\ref{app:summariazation:human_agreement},~\\ref{app:summariazation:human_tukey} show averaged results, annotator agreement, and the results of statistical significance tests to determine which models output better generations when rated by humans.\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=.48\\linewidth]{figures\/summarization_human_study\/summarization_human1.png}\n    \\includegraphics[width=.48\\linewidth]{figures\/summarization_human_study\/summarization_human3.png}\n    \\caption{Instructions and interface for the summarization task.}\n    \\label{fig:description_interface_summarization}\n\\end{figure*}\n\n\n\\subsubsection{Qualitative Analysis}\n\\label{app:sec:summ_qual_analysis}\nWe show sample generations from each of the algorithms for three randomly picked prompts below.\n\n\\begin{lstlisting}\nSample 1\nPrompt: Manchester City are confident UEFA's punishment for breaching financial fairplay regulations will be lifted this summer which would allow them to bid for stellar names like Raheem Sterling, Gareth Bale, Kevin de Bruyne and  Ross Barkley. City boss Manuel Pellegrini has been hampered over the past year by UEFA restricting them to a net transfer spend of 49million in each window and keeping the club's overall wage bill to its current level of 205million-a-year. UEFA's settlement with City published in May stated those penalties would remain in place until the end of the 2015\/16 season but the club's latest financial figures showed drastically-reduced losses of 23million which they feel proves they are now compliant with FPP regulations. Manuel Pellegrini is hoping that the financial  restrictions imposed by UEFA for a breach of FFP rules will be lifted at the end of this season . Manchester City have been limited to a net spend of 49million in the last two transfer windows - they spent 25m bringing Wilfried Bony in from Swansea in January . Ahead of Monday night's trip to Crystal Palace, Pellegrini was certainly talking like a man excited at the prospect of signing 'crack' players this summer. 'I think that next season we don't have any restrictions so we will be in the same position that all the other English clubs have,' said Pellegrini. 'It's important. You have so many strong teams here in England and in Champions League, you can not allow them to keep the advantage every year; having less players  to put in your squad or spending less money. We spend money, of course we always spend money, but they spent more.' Manchester United, Barcelona, Liverpool and Arsenal have all paid more in transfer fees in the past 12 months than City who were traditionally Europe's biggest spenders after the club was taken over by Abu Dhabi owners in 2008. Uefa also ordered City to play with a reduced squad from 25 players to 21 in the Champions League this season and while that restriction has now ended, any time reduction in the penalties on spending and wages is more controversial. Arsenal have paid more in transfer fees than City in the last 12 months, including 30m on Alexis Sanchez . The document published last May by UEFA's Club Financial Control Body investigative chamber explicitly said City's financial penalties would run for two seasons  at least and there has been no official deviation from that decision. \nThe published statement said at the time: 'Manchester City agrees to significantly limit spending in the transfer market for the seasons 2014\/15 and 2015\/16. It means City will have to argue their case with Uefa that as they have been financially compliant over the past year, they deserve to be free of restrictions moving forward. They have successfully argued their case with UEFA before. Last summer they persuaded the governing body to allow them to bypass the normal quota of eight homegrown players as their Champions League squad had been reduced. Eliaquim Mangala joined the champions from Porto for 32m last summer . The reigning Premier League champions have only paid money for Fernando, Willie Caballero, Eliaquim Mangala and Wilfried Bony in the last two transfer windows and that was part-paid by the sales of Javi Garcia and Jack Rodwell. Pellegrini admitted they weren't in a position to compete with neighbours United for established world stars like Angel di Maria because of the FFP penalties. It has cost City on the pitch. Even if they win their remaining eight Premier League fixtures their points tally will be lower than last season and they crashed out once again at the last-16 stage of the Champions League. Pellegrini and director of football Txiki Begiristain both accept the squad needs replenishing and they want to be in the market for the top available players. Manchester City have been heavily linked with a move for former Chelsea midfielder Kevin de Bruyne . The club also want to be involved in any race for Liverpool contract rebel Raheem Sterling . They have been strongly linked with former Chelsea winger Kevin de Bruyne who has been one of the best players in the Bundesliga with Wolfsburg. And if Liverpool do allow Sterling to leave, City want to be involved in the chase for one of England's most outstanding talents to help fit in with quotas for homegrown players. 'If you are the manager or a player of a big tem, you are always open to criticism because everyone expects a high level of performance,' he said. 'The major issue with our team always seems to be the money but this season I think we were the team that spent less than any other team. That is the weird thing. 'For this club trying to be a big team in so few years, maybe we are paying the cost for that.' Since the transfer penalties were introduced, City have spent 91million on players in the last two windows (Net spend 68million). That compares to Manchester United 145.5million, Liverpool 113million, Arsenal 92.5million and Chelsea 82.6million. Over the same time period Barcelona spent 118.3million on players and Real Madrid 81.2million though they also broke the world transfer record for Gareth Bale the previous summer.\nZero-Shot: manuel Pellegrini hoping UEFA's punishment for breaching financial fairplay regulations will be lifted this summer.  city have been restricted to a net transfer spend of 49million in each window. a total of 23million has been lost in the last year.\nReferences:  Manchester City have been restricted to a net transfer spend of 49m .Club's also had to keep overall wage bill to its  current level of 205m .Punishments imposed by UEFA for breaching Financial Fair Play rules .The spending restrictions were set for this season and the next one .But City are confident they will be lifted early after their compliance .\nPPO: Manuel Pellegrini is hoping UEFA's punishment for breaching financial fairplay rules will be lifted this summer. Manchester City have been limited to a net spend of 49million in each transfer window. the club also keep the club's overall wage bill to its current level of 205million-a-year.\nNLPO:  Manchester City are confident UEFA's punishment for breaching financial fairplay regulations will be lifted this summer. that would allow them to bid for stellar names like Raheem Sterling, Kevin de Bruyne and Ross Barkley.\nSupervised: Manchester City have been limited to a net transfer spend of 49million in each window. City's overall wage bill is 205m-a-year due to a breach of UEFA rules. City are confident the financial restrictions will be lifted at the end of the current season in order to bid for stars such as Raheem Sterling. Manuel Pellegrini is hoping that the financial restrictions will be lifted at the end of the 2015\/16 season\nSupervised + PPO: Manchester City have been restricted to a net spend of 49million in each window and kept their overall wage bill at 205m. Manuel Pellegrini believes these restrictions will be lifted at the end of the season. Arsenal have spent more in transfer fees than City in the last 12 months, including 30m on Alexis Sanchez.\nSupervised + NLPO: Manchester City are confident UEFA's punishment for a breach of financial fairplay regulations will be lifted this summer. The club have been restricted to a net transfer spend of 49million over the past two window. They aim to bid for stars like Raheem Sterling, Gareth Bale and Kevin de Bruyne. City lost 23m on transfers in the last 12 months due to financial restrictions imposed by UEFA for breach \n\nSample 2\nPrompt: (CNN)Larry Johnson remembers the fear and feeling of helplessness from being on the SkyWest Airlines flight that made an emergency landing in Buffalo, New York. \"I thought we were done,\" he said Thursday, one day later. \"There was no doubt in my mind that we weren't going to land.\" Johnson was flying with his brother, his girlfriend and his 8-month-old son when he says a flight attendant came over the speaker asking for someone who was medically trained to help with a sick passenger. Minutes later, Johnson says, the attendant announced there was a pressurization problem and told passengers to prepare for the emergency landing. \"You're going to feel dizzy and woozy and lightheaded, and you're not going to be able to breathe,\" Johnson recalled the flight attendant saying. The Federal Aviation Administration on Wednesday initially reported a pressurization problem with SkyWest Flight 5622, and said it would investigate. It later issued a statement that did not reference any pressurization issues. \"It was like being trapped and you couldn't do anything. You have no control over it. Your body just felt pressured,\" Johnson said. Johnson said his girlfriend was seeing spots and that \"when I was blinking, it was really bright. I could barely see.\" According to Marissa Snow, spokeswoman for SkyWest, three passengers reported a loss of consciousness while on the flight. Fifteen passengers were examined by medical personnel at the Buffalo airport, but no one was transported to the hospital, she said. The spokeswoman said that maintenance personnel found no indication of a pressurization problem with the aircraft, an Embraer E170, and that the airline continues to investigate the cause. An official with the National Transportation Safety Board told CNN that the agency is in communication with the FAA and SkyWest to gather information on the incident to better understand what took place.\nMary Cunningham was the nurse on board who attended to the first sick passenger. \"There was a woman, very lethargic, very nauseous, turning gray, just not looking good,\" Cunningham said. After administering oxygen to the woman, Cunningham, a nurse at Yale-New Haven Hospital in Connecticut, returned to her seat. Soon after, Cunningham said, she was called up again by the flight attendant after another women near the first started to lose consciousness. \"It was something that was affecting oxygen in that area,\" which was near the middle of the plane, she said. Cunningham said she had to return to her seat after feeling lightheaded. There were 84 passengers on board the plane, including nine crew members. Flight 5622 was originally scheduled to fly from Chicago to Hartford, Connecticut. The plane descended 28,000 feet in three minutes. \"It would feel like a roller coaster -- when you're coming over the top and you're going down,\" CNN aviation analyst Mary Schiavo said, describing how such a descent would feel. \"You know that these pilots knew they were in a very grave and very serious situation.\" Johnson, flying to visit his parents with their only grandchild, was glad that roller coaster ride ended safely. \"It was a shaky ride. It wasn't anything I would like to experience again,\" he said. CNN's Jean Casarez, Josh Gaynor, Dana Ford and Stephanie Gallman contributed to this report.\nZero-Shot: \"you're going to feel dizzy and woozy and lightheaded,\" Johnson says. \"it was like being trapped and you couldn't do anything,\" Johnson says.  \"the plane was trying to get back on the ground,\" passenger says of fear.\nReferences: Three passengers report a loss of consciousness on SkyWest flight .But officials say there is no evidence of a pressurization problem .\nPPO: SkyWest flight 5312 made emergency landing in Buffalo, new york. passengers were told they were not going to be able to breathe,  passenger recalls attendant. the faa initially reported a pressurization problem with the flight 5622.\nNLPO: SkyWest flight 5622 made emergency landing in buffalo, new york.Larry Johnson says flight attendant announced there was a pressurization problem. three passengers reported loss of consciousness while on flight; no one transported to hospital.\nSupervised: The FAA says it is investigating the cause of the emergency landing. SkyWest Flight 5622 made an emergency landing in Buffalo, New York.  The flight was carrying a family of three people. The flight attendant told the passengers to prepare for the landing.\nSupervised + PPO: \"I thought we were done,\" the father of one passenger says. The flight attendant announced there was a pressurization problem.  The flight made an emergency landing in Buffalo, New York. Three passengers reported a loss of consciousness while on the flight.\nSupervised + NLPO: SkyWest Airlines flight made an emergency landing in Buffalo, New York, on Thursday. \"There was no doubt in my mind  that we weren't going to land,\" passenger says. \"You're going to feel dizzy and woozy and lightheaded...you're not going to be able to breathe,\" he says.\"\n\nSample 3\nPrompt: For Inverness the latest chapter in their remarkable story featured not one, but two, plot-changing twists. One featured penalty Celtic didn't get for 2-0 in which Josh Meekings handball should have also led to a sending off. The other the spot kick they did, followed by a red card for Craig Gordon. 'I've not seen it yet, but going by the reaction of the Celtic players we got away with a penalty and a sending off and that was probably the turning point in the game,' acknowledged Caley manager John Hughes after. Inverness's Josh Meekings appears to get away with a handball on the line in their win over Celtic . Caley boss John Hughes says the break, which could have meant a penalty and red card, was a turning point . 'I've not spoken to Josh. I haven't seen it - but going by the media it was definitely a hand ball. We look at the referee behind the line and all that and I know Ronny will feel aggrieved - because I certainly would. 'But it's part and parcel of football and you need a wee bit of luck to beat Celtic. 'This was their biggest game of the season because they will go on and win the league and if they had beaten us today there was a good chance they would have gone on and won the Scottish Cup. 'But when Marley Watkins was clipped by Craig Gordon and they were down to 10 men that was advantage Inverness. 'We weren't going to give Celtic the ball back, they had to come and get it and we had to be patient. 'When big Edward put us into the lead we thought it was going to be our day on the back of things that had happened. 'Celtic equalised with another free kick but it's typical of Inverness that we don't do anything easy. 'We do it the hard way and we came up with the winner through David Raven.' Hughes hauled Raven, his Scouse defender, from his backside as extra-time beckoned. Offended by the sight of one of his players resting he had a message to impart. Caley players celebrate after upsetting Celtic in a Scottish Cup semi-final 3-2 thriller . Celtic, depleted by games and absentees, were virtually on their knees after a relentless programme of midweek games. In last season's League Cup Final Inverness had been passive and unambitious prior to losing on penalties. This was no time to repeat the mistake. 'I tried to emphasise to the players they would never have a better time to go on and beat Celtic, down to 10 men in the semi final of a cup. We needed to go for it,' Hughes said. 'Before Raven scored at the back post I was looking to change it. \nI was going to bring on another winger, Aaron Doran, and put him in the full-back position over on the right, but more advanced so he could take their left back on. Thankfully I didn't do that and David Raven came up with the goal. Virgil Van Dijk (centre) fired Celtic into an early lead with a superb free-kick in the 18th minute . 'I didn't realise this is the first time the club have been in the final of the Scottish Cup and that's a remarkable achievement given it was only formed 20 years ago. 'It is a great story isn't it? It's an absolutely fantastic story. It is 20 odd years since the amalgamation. We are a small provincial club up there in the Highlands. 'We have lost a real inspirational skipper in Richie Foran right from the start of the season. He has never played. We have had to adjust to that. 'We had to sell Billy McKay, our top goalscorer, at Christmas. We have had to go again and adjust. I am a very humble guy and I am grateful and thankful that injuries have never caught up with us.'  There is remarkable irony in the fact Falkirk will be the opponents for the final. A former Bairns captain, he was manager of the club in 2009 when they lost to Rangers at Hampden. Former Falkirk captain and manager John Hughes will take on his former club in the final . 'I had a lot of great times at Falkirk. So much so that it is possibly my favourite time in my playing career. I am still friendly with an awful lot of the characters who were in that dressing room. Neil Oliver is a good friend of mine from my Falkirk days. He comes along and sits on the bench and gives me a wee hand out. 'That is the spirit that we had at that club. I have met some great guys. Brian Rice, my ex-coach, Davie Weir, Ian McCall, the list is endless. I was just talking the other day about that day at Falkirk. There are times even now when I see. 'I have a real special, special feel for the place. I am not kidded Falkirk is a right good side, trust me on that. No side goes to Tynecastle and beats Hearts without being a good side. No side beats Hibs the way they were playing without being a good side. You might say we are favourites I say it is even stevens and I am just looking forward to when the days comes.' David Raven (second right) celebrates with his Inverness team-mates following his extra-time winner . An Inverness and Falkirk final would not have been the choice of sponsors  William Hill, you suspect. Vast swathes of Hampden's west stand were empty, a 12.15 kick-off for the benefit of broadcasters proving unhelpful . There will be no ticket rows this time. Not over a final unlikely to raise a crowd much higher than 35,000 in a stadium of 52,000. The contract with Hampden plc is signed, with no question of moving it. If the bitter amalgamation of 21 years ago produced a fine football club it also disenfranchised many of those who followed the old ones. 'I hope this encourages the next generation to follow Inverness,' added Hughes. 'In the peer groups it is still Celtic and Rangers they follow but I hope the next ones will want to support their local club. 'If we can add another few hundred or thousand to the support then we are playing our part.\nZero-Shot: inverness beat Celtic 2-0 in their opening game of the season. Josh Meekings was the subject of a handball after the break. the break could have meant a penalty and a red card for Craig Gordon.\nReferences: Josh Meekings handball should have been a penalty and a red card .If the penalty was awarded and converted Celtic would have gone to 2-0 .Caley manager John Hughes admitted his side were fortunate in win .Virgil van Dijk scored Celtic's opener with a superb free-kick .Celtic keeper Craig Gordon was sent off early in the second half .Greg Tansey, Edward Ofere and finally Daven Raven scored for Caley .\nPPO: Josh Meekings handball could have meant sending off for Celtic in extra-time win. John Hughes says the break could have meant an penalty and red card. Caley boss also admits he would feel aggrieved had Celtic won the championship.\nNLPO: Caley beat Celtic 2-0 in their Scottish Cup clash. Josh Meekings handball could have led to a penalty and red card. John Hughes says the break was probably the turning point in the game. defender does not believe defensive striker can get away with being shown the yellow card - despite playing in same half we never scored.\nSupervised: Inverness beat Celtic 2-0 in their Scottish Cup clash. Josh Meekings appeared to get away with a handball on the line. Another player was sent off for a spot kick which ended in a red card. Caley manager John Hughes says the break could have meant a penalty. Hughes says he 'would feel aggrieved' after the break.\nSupervised + PPO: Caley beat Celtic 1-0 in their Scottish Cup encounter at the Etihad. Josh Meekings' handball on the line should have also seen a sending off.  But Craig Gordon got away with a spot kick and sending off. John Hughes said it was a turning point in the game for Inverness.\nSupervised + NLPO: Josh Meekings appears to get away with a handball after Inverness's win. Inverness were down to 10 men when Craig Gordon was sent off.  John Hughes says the break could have meant a penalty and red card. Hughes says Celtic will go on and win the Scottish Cup with win \n\\end{lstlisting}\n\\subsubsection{Setup} ToTTo \\citep{parikh2020totto} is a controlled table-to-text generation task in which the goal is to produce one-sentence description of highlighted table cells.\nFor training RL methods, we consider 5 different reward functions: BLEU, SacreBLEU, METEOR, PARENT and a combination of Meteor and PARENT. We chose T5 as our base LM here too, as they are more suitable for structure to text tasks. For decoding, we use beam search during inference and for generating rollouts, we use top k sampling.  Other implementation details are captured in Table~\\ref{tbl:totto_gen_hyperparams}.\n\n\\begin{table*}[ht!]\n\\centering\n\\footnotesize\n\\resizebox{0.5\\textwidth}{!}{\n\\begin{tabular}{ll}\n\\toprule\n\\textbf{Model Params}\n& \\multicolumn{1}{c}{\\textbf{value}} \\\\ \n\\cmidrule{1-2}\nsupervised & batch size: $8$\\\\\n& epochs: $4$ \\\\\n& learning rate: $0.0001$ \\\\\n& learning rate scheduler: constant with warm up \\\\\n& weight decay: $0.1$ \\\\\n\\cmidrule{1-2}\nppo\/\\textcolor{forestgreen2}{nlpo} & steps per update: $2560$\\\\\n    & total number of steps: $256000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$\\\\\n    & learning rate: $0.000002$ \\\\\n    & entropy coefficient: $0.0$ \\\\\n    & initial kl coeff: $0.001$ \\\\\n    & target kl: $2.0$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & rollouts top k : $0$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.9$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: $20$} \\\\\n\\cmidrule{1-2}\nsupervised+ ppo (or \\textcolor{forestgreen2}{nlpo}) & steps per update:$2560$\\\\\n    & total number of steps: $256000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$ \\\\\n    & learning rate: $0.0000005$ \\\\\n    & entropy coefficient: $0.0$ \\\\\n    & initial kl coeff: $0.01$ \\\\\n    & target kl: $0.2$\\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & rollouts top k : $50$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.9$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: $20$} \\\\\n\\cmidrule{1-2}\n\\cmidrule{1-2}\ndecoding & num beams: $5$ \\\\\n& min length: $10$ \\\\\n& max new tokens: $50$\\\\\n\n\\cmidrule{1-2}\ntokenizer & padding side: left\\\\\n& truncation side: right\\\\\n& max length: 512 \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{ToTTO Hyperparams}: Table shows a list of all hyper-parameters and their settings}\n       \\label{tbl:totto_gen_hyperparams}\n\\end{table*}\n\n\\subsubsection{Results and Discussion} Tables \\ref{tbl:totto_val_scores}, \\ref{tbl:totto_test_scores}  presents our benchmarking results with 5 reward functions along with supervised baseline performances on dev and test sets respectively. Similar to other tasks, our main finding is that warm-started initial policies are crucial for learning to generate descriptions from highlighted cells. Without warm-start, policies suffer from reward hacking and resulting in sub-optimal solutions despite application of task-specific metrics such as PARENT etc. We find that Supervised+NLPO method outperforms all models on ToTTo leaderboard in terms of PARENT metric.\n\n\\begin{table*}[h]\n       \\centering\n       \\resizebox{1.0\\textwidth}{!}{\n       \\begin{tabular}{@{}ccccccccccccc@{}}\n       \\toprule\n         Tasks \n         & \\multicolumn{3}{c}{\\textbf{\\_}} \n         & \\multicolumn{6}{c}{\\textbf{Lexical and Semantic Metrics}} \n         & \\multicolumn{3}{c}{\\textbf{Factual Consistency}}\n         \\\\\n         & Alg & LM & Reward function & \\multicolumn{3}{c}{SacreBleu} & \\multicolumn{3}{c}{BLEURT}\n         & \\multicolumn{3}{c}{PARENT} \\\\\n         \\cmidrule(lr){5-7} \\cmidrule(lr){8-10} \\cmidrule(lr){11-13} \\\\\n         & & &  & Overall & Overlap & Non-Overlap & Overall & Overlap & Non-Overlap & Overall & Overlap & Non-Overlap\\\\\n         \\cmidrule(lr){2-2}  \\cmidrule(lr){3-3} \\cmidrule(lr){4-4} \\cmidrule(lr){5-5}\n         \\cmidrule(lr){6-6} \\cmidrule(lr){7-7} \\cmidrule(lr){8-8} \\cmidrule(lr){9-9}\n         \\cmidrule(lr){10-10}\n         \\cmidrule(lr){11-11}\n         \\cmidrule(lr){12-12}\n         \\cmidrule(lr){13-13}\n         \\multirow{22}{*}{ToTTo} & Zero-Shot & T5 & &  0.036 & 0.040 & 0.032 & -1.392 & -1.387 & -1.397 & 0.116 & 0.119 & 0.112\\\\\n         \\cmidrule{2-13}\n         & PPO & T5 & bleu & 0.065 & 0.067 & 0.063 & -1.074 & -1.045 & -1.098 & 0.246 & 0.246 & 0.244 \\\\\n         &  & T5 & sacrebleu & 0.086 & 0.090 & 0.083 & -0.979 & -0.955 & -1.003 & 0.293 & 0.292 & 0.294\\\\\n         & & T5 & meteor & 0.144 & 0.155 & 0.132 & -0.769 & -0.713 & -0.826 & 0.356 & 0.361 & 0.351\\\\\n         & & T5 & parent & 0.146 & 0.153 & 0.128 & -0.721 & -0.688 & -0.753 & 0.336 & 0.335 & 0.339\\\\\n         & & T5 & meteor + parent & 0.161 & 0.169 & 0.152 & -0.891 & -0.861 & -0.922 & 0.345 & 0.342 & 0.348 \\\\\n         \n         \\cmidrule{2-13}\n         & NLPO & T5 & bleu & 0.062 & 0.065 & 0.059 & -1.077 & -1.057 & -1.097 & 0.235 & 0.236 & 0.233 \\\\\n         &  & T5 & sacrebleu & 0.085 & 0.088 & 0.083 & -0.945 & -0.917 & -0.972 & 0.314 & 0.315 & 0.313\\\\\n         & & T5 & meteor & 0.102 & 0.108 & 0.097 & -1.044 & -1.009 & -1.079 & 0.329 & 0.328 & 0.330\\\\\n          & & T5 & parent & 0.159 & 0.166 & 0.152 & -0.710 & -0.675 & -0.745 & 0.357 & 0.351 & 0.363\\\\\n         & & T5 & meteor + parent & \\textbf{0.166} & \\textbf{0.175} & \\textbf{0.158} & \\textbf{-0.704} & \\textbf{-0.668} & \\textbf{-0.740} & \\textbf{0.365} & \\textbf{0.362} & \\textbf{0.368}\\\\\n         \n         \\cmidrule{2-13}\n         \n         & Supervised & T5 & & 0.457 & 0.535 & 0.377 & 0.204 & 0.327 & 0.081 & 0.583 & 0.631 & 0.534  \n         \\\\\n         \n         \\cmidrule{2-13}\n         & Supervised + PPO & T5 & bleu & 0.473 & 0.548 & 0.395 & 0.200 & 0.323 & 0.078 & 0.590 & 0.638 & 0.542 \\\\\n          & & T5 & sacrebleu & 0.474 & 0.557 & 0.389 & \\textbf{0.209} & \\textbf{0.340} & 0.077 & 0.573 & 0.620 & 0.525 \\\\\n          & & T5 & meteor & 0.468 & 0.541 & 0.392 & 0.203 & 0.325 & \\textbf{0.082} & 0.590 & 0.638 & 0.542\\\\\n          & & T5 & parent &  0.469 & 0.547 & 0.388 & 0.175 & 0.300 & 0.050 & 0.595 & 0.641 & 0.549\\\\\n         & & T5 & meteor + parent & 0.473 & 0.547 & 0.392 & 0.192 & 0.314 & 0.069 & 0.595 & 0.642 & 0.549\\\\\n         \n         \\cmidrule{2-13}\n         & Supervised + NLPO & T5 & bleu & \\textbf{0.475} & 0.548 & \\textbf{0.399} & 0.208 & 0.330 & 0.085 & 0.593 & 0.639 & 0.546\\\\\n         &  & T5 & sacrebleu & 0.475 & 0.557 & 0.392 & 0.208 & 0.335 & 0.081 & 0.577 & 0.625 & 0.529\\\\\n         & & T5 & meteor & 0.468 & 0.541 & 0.392 & 0.201 & 0.322 & 0.079 & 0.594 & 0.641 & 0.546\\\\\n         & & T5 & parent & 0.474 & \\textbf{0.550} & 0.392 & 0.192 & 0.315 & 0.068 & \\textbf{0.596} & \\textbf{0.643} & \\textbf{0.550} \\\\\n         & & T5 & meteor + parent & 0.471 & 0.546 & 0.393 & 0.204 & 0.326 & 0.081 & 0.592 & 0.640 & 0.544\\\\\n         \\\\\n         \\bottomrule\n        \n       \\end{tabular}\n       }\n       \\caption{\\textbf{ToTTo test evaluation}: Table shows lexical, semantic and factual correctness metric scores of algorithms with different reward functions on hold-out test set. Without supervised pre-training, both PPO and NLPO results in sub-optimal solutions, with NLPO better than PPO. With supervised pre-training, PPO and NLPO achieve better scores across all metrics showing RL fine-tuning is beneficial. Most importantly, RL fine-tuned models produce more factually consistent text as seen in higher PARENT scores. Another observation, fine-tuning with a task-specific metric PARENT is better than training on task-agnostic lexical rewards}\n       \\label{tbl:totto_test_scores}\n    \\end{table*}\n\n\n\\begin{landscape}\n\\begin{table*}[h]\n       \\centering\n       \\resizebox{1.5\\textwidth}{!}{\n       \\begin{tabular}{@{}cccccccccccccccccccccccc@{}}\n       \\toprule\n         Tasks \n         & \\multicolumn{3}{c}{\\textbf{\\_}} \n         & \\multicolumn{9}{c}{\\textbf{Lexical and Semantic Metrics}} \n         & \\multicolumn{3}{c}{\\textbf{Factual Consistency}} \n         & \\multicolumn{8}{c}{\\textbf{Diversity Metrics}}\n         \\\\\n         & Alg & LM & Reward function & Rouge-1 & Rouge-2 & Rouge-L & Rouge-LSum & Meteor & BertScore & \\multicolumn{3}{c}{SacreBleu}\n         & \\multicolumn{3}{c}{PARENT} \\\\\n         \\cmidrule(lr){11-13} \\cmidrule(lr){14-16} \\\\\n         & & & & & & & &  & & Overall & Overlap & Non-Overlap & Overall & Overlap & Non-Overlap & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ & Mean Output Length\\\\\n         \n         \\cmidrule(lr){2-2}  \\cmidrule(lr){3-3} \\cmidrule(lr){4-4} \\cmidrule(lr){5-5}\n         \\cmidrule(lr){6-6} \\cmidrule(lr){7-7} \\cmidrule(lr){8-8} \\cmidrule(lr){9-9}\n         \\cmidrule(lr){10-10} \\cmidrule(lr){11-11} \\cmidrule(lr){12-12}\n         \\cmidrule(lr){13-13} \\cmidrule(lr){14-14} \\cmidrule(lr){15-15}\n         \\cmidrule(lr){16-16} \\cmidrule(lr){17-17} \\cmidrule(lr){18-18}\n         \\cmidrule(lr){19-19} \\cmidrule(lr){20-20} \\cmidrule(lr){21-21}\n         \\cmidrule(lr){22-22} \\cmidrule(lr){23-23} \\cmidrule(lr){24-24}\n         \n         \\multirow{22}{*}{ToTTo} & Zero-Shot & T5 & & 0.131 & 0.055 & 0.127 & 0.127 & 0.057 & 0.805 & 0.038 & 0.042 & 0.034 & 0.118 & 0.119 & 0.116 & 0.428 & 0.084 & 0.238 & 6.703 & 9.933 & 8387 & 26490 & 19.964\n         \\\\\n         \\cmidrule{2-24}\n         & Supervised & T5 & & 0.410 & 0.279 & 0.388 & 0.388 & 0.223 & 0.953 & 0.458 & 0.533 & 0.387 & 0.586 & 0.633 & 0.540 & 0.715 & 0.162 & 0.511 & 9.995 & 14.468 & 15168 & 54706 & 17.791\n         \\\\\n         \\cmidrule{2-24}\n         & PPO & T5 & bleu & 0.274 & 0.138 & 0.249 & 0.249 & 0.139 & 0.844 & 0.068 & 0.071 & 0.066 & 0.251 & 0.250 & 0.251 & 0.403 & 0.091 & 0.308 & 10.659 & 14.511 & 7536 & 34232 & 28.545\\\\\n         &  & T5 & sacrebleu & {0.341} & {0.166} & {0.300} & {0.300} & 0.165 & 0.858 & 0.09 & 0.094 & 0.086 & 0.300 & 0.299 & 0.300 & 0.469 & 0.121 & 0.407 & 11.071 & 14.880 & 10138 & 48195 & 26.612 \\\\\n         \\rowcolor{lightgray}\n         & & T5 & meteor & 0.322 & 0.157 & 0.286 & 0.286 & {0.173} & 0.888 & 0.147 & 0.163 & 0.133 & {0.358} & {0.367} & 0.350 & 0.625 & 0.136 & 0.482 & 10.189 & 14.910 & 12346 & 54925 & 21.484\\\\\n         & & T5 & parent & 0.268 & 0.125 & 0.251 & 0.251 & 0.119 & {0.890} & 0.150 & 0.158 & 0.143 & 0.337 & 0.332 & 0.342 & 0.764 & 0.202 & 0.646 & 11.068 & 14.988 & 13068 & 50313 & 13.035\\\\\n         & & T5 & meteor + parent & 0.266 & 0.128 & 0.251 & 0.251 & 0.130 & 0.886 & {0.165} & 0.175 & {0.155} & 0.348 & 0.346 & {0.350} & 0.702 & 0.181 & 0.594 & 10.096 & 14.432 & 14422 & 55770 & 15.354\\\\\n         \n         \\cmidrule{2-24}\n         & NLPO & T5 & bleu & 0.267 & 0.134 & 0.24 & 0.24 & 0.137 & 0.84 & 0.068 & 0.071 & 0.065 & 0.238 & 0.239 & 0.237 & 0.448 & 0.1 & 0.359 & 11.259 & 14.623 & 9029 & 47209 & 28.472 \\\\\n         \\rowcolor{lightgray}\n         &  & T5 & sacrebleu & 0.341 & 0.168 & 0.297 & 0.297 & 0.183 & 0.863 & 0.089 & 0.093 & 0.085 & 0.32 & 0.324 & 0.317 & 0.494 & 0.111 & 0.373 & 11.007 & 15.032 & 9455 & 43379 & 27.977 \\\\\n         & & T5 & meteor & 0.322 & 0.157 & 0.286 & 0.286 & 0.173 & 0.888 & 0.147 & 0.163 & 0.133 & 0.358 & 0.367 & 0.350 & 0.625 & 0.136 & 0.482 & 10.189 & 14.910 & 12346 & 54925 & 21.484\\\\\n         & & T5 & parent & 0.283 & 0.132 & 0.264 & 0.264 & 0.133 & 0.894 & 0.163 & 0.174 & 0.153 & 0.36 & 0.357 & 0.364 & 0.824 & 0.223 & 0.691 & 11.493 & 15.127 & 14344 & 55542 & 14.204\\\\\n         & & T5 & meteor + parent & 0.299 & 0.14 & 0.276 & 0.276 & 0.142 & 0.896 & 0.171 & 0.181 & 0.161 & 0.369 & 0.365 & 0.372 & 0.779 & 0.214 & 0.674 & 11.072 & 15.275 & 14939 & 58737 & 15.141\\\\\n         \n         \\cmidrule{2-24}\n          \\rowcolor{lightgray}\n         & Supervised + PPO & T5 & bleu & 0.408 & 0.283 & 0.388 & 0.388 & 0.222 & 0.954 & \n          0.477 & 0.549 & 0.405 & 0.596 & 0.644 & 0.550 & 0.722 & 0.167 & 0.525 & 10.080 & 14.524 & 15203 & 54724 & 17.296\\\\\n          & & T5 & sacrebleu &  0.395 & 0.275 & 0.378 & 0.378 & 0.211 & 0.955 & 0.477 & 0.554 & 0.401 & 0.577 & 0.621 & 0.535 & 0.728 & 0.174 & 0.539 & 10.086 & 14.518 & 14846 & 52327 & 16.063 \\\\\n          & & T5 & meteor & 0.410 & 0.282 & 0.389 & 0.389 & 0.223 & 0.954 &  0.469 & 0.540 & 0.398 & 0.593 & 0.642 & 0.547 & 0.718 & 0.165 & 0.516 & 10.037 & 14.467 & 15182.0 & 54446 & 17.542\\\\\n          & & T5 & parent & 0.401 & 0.277 & 0.382 & 0.382 & 0.215 & 0.953 & 0.470 & 0.543 & 0.394 & 0.598 & 0.647 & 0.550 & 0.732 & 0.174 & 0.545 & 10.209 & 14.660 & 15379.0 & 55421 & 16.826 \\\\\n         & & T5 & meteor + parent & 0.406 & 0.281 & 0.386 & 0.387 & 0.220 & 0.954 & 0.473 & 0.544 & 0.399 & 0.600 & 0.648 & 0.553 & 0.727 & 0.170 & 0.532 & 10.143 & 14.586 & 15330 & 55211 & 17.185\\\\\n         \n         \n         \\cmidrule{2-24}\n         & Supervised + NLPO & T5 & bleu & 0.410 & 0.283 & 0.388 & 0.388 & 0.222 & 0.954 & 0.476 & 0.548 & 0.404 & 0.597 & 0.644 & 0.552 & 0.721 & 0.167 & 0.524 & 10.077 & 14.532 & 15213 & 54948 & 17.408\\\\\n         &  & T5 & sacrebleu & 0.397 & 0.276 & 0.38 & 0.38 & 0.214 & 0.955 & 0.477 & 0.555 & 0.401 & 0.581 & 0.628 & 0.535 & 0.729 & 0.174 & 0.54 & 10.124 & 14.544 & 14940 & 52986 & 16.334 \\\\\n         \\rowcolor{lightgray}\n         & & T5 & meteor & 0.411 & 0.283 & 0.389 & 0.39 & 0.224 & 0.954 & 0.474 & 0.547 & 0.403 & 0.6 & 0.649 & 0.554 & 0.727 & 0.171 & 0.536 & 10.156 & 14.612 & 15341 & 55292 & 17.637 \\\\\n         & & T5 & parent & 0.405 & 0.28 & 0.386 & 0.386 & 0.219 & 0.954 & 0.469 & 0.541 & 0.398 & 0.598 & 0.645 & 0.552 & 0.716 & 0.165 & 0.519 & 10.019 & 14.5 & 15218 & 54793 & 17.095 \\\\\n         & & T5 & meteor + parent & 0.405 & 0.28 & 0.386 & 0.386 & 0.219 & 0.954 & 0.474 & 0.547 & 0.398 & 0.598 & 0.646 & 0.552 & 0.727 & 0.171 & 0.536 & 10.156 & 14.612 & 15341 & 55292 & 17.095 \\\\\n         \n         \n       \\bottomrule\n       \\end{tabular}\n       }\n       \\caption{\\textbf{ToTTo dev evaluation}: Table shows lexical, semantic and factual correctness metric scores of algorithms with different reward functions on dev set. Without supervised pre-training, both PPO and NLPO results in sub-optimal solutions, with NLPO better than PPO. With supervised pre-training, PPO and NLPO achieve better scores across all metrics showing RL fine-tuning is beneficial. Most importantly, RL fine-tuned models produce more factually correct text as seen in higher PARENT scores. Another observation, fine-tuning with a task-specific metric PARENT is better than training just on task-agnostic lexical metrics}\n       \\label{tbl:totto_val_scores}\n    \\end{table*}\n    \\end{landscape}\n\n\n\n\n\\begin{table}[]\n\\centering\n\\footnotesize\n\\begin{tabular}{|l|r|rrr|rrr|}\n\\multicolumn{1}{|c|}{\\multirow{2}{*}{\\textbf{Algorithm}}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{Unique N}}} & \\multicolumn{3}{c|}{\\textbf{Coherence}}                                                                        & \\multicolumn{3}{c|}{\\textbf{Correctness}}                                                                      \\\\\n\\multicolumn{1}{|c|}{}                                    & \\multicolumn{1}{c|}{}                                   & \\multicolumn{1}{c|}{\\textbf{Value}} & \\multicolumn{1}{c|}{\\textbf{Alpha}} & \\multicolumn{1}{c|}{\\textbf{Skew}} & \\multicolumn{1}{c|}{\\textbf{Value}} & \\multicolumn{1}{c|}{\\textbf{Alpha}} & \\multicolumn{1}{c|}{\\textbf{Skew}} \\\\ \\hline\nZero Shot                                                 & 25                                                      & \\multicolumn{1}{r|}{1.63}           & \\multicolumn{1}{r|}{0.718}          & 1.642                              & \\multicolumn{1}{r|}{1.93}           & \\multicolumn{1}{r|}{0.503}          & 1.946                              \\\\\nPPO+Supervised                                            & 24                                                      & \\multicolumn{1}{r|}{4.57}           & \\multicolumn{1}{r|}{0.221}          & 4.579                              & \\multicolumn{1}{r|}{4.48}           & \\multicolumn{1}{r|}{0.098}          & 4.483                              \\\\\nPPO                                                       & 26                                                      & \\multicolumn{1}{r|}{2.75}           & \\multicolumn{1}{r|}{0.427}          & 2.753                              & \\multicolumn{1}{r|}{3.23}           & \\multicolumn{1}{r|}{0.214}          & 3.227                              \\\\\nNLPO                                                      & 28                                                      & \\multicolumn{1}{r|}{2.25}           & \\multicolumn{1}{r|}{0.401}          & 2.247                              & \\multicolumn{1}{r|}{2.61}           & \\multicolumn{1}{r|}{0.419}          & 2.613                              \\\\\nSupervised                                                & 24                                                      & \\multicolumn{1}{r|}{\\textbf{4.59}}  & \\multicolumn{1}{r|}{0.173}          & 4.592                              & \\multicolumn{1}{r|}{4.54}           & \\multicolumn{1}{r|}{0.189}          & 4.537                              \\\\\nNLPO+Supervised                                           & 26                                                      & \\multicolumn{1}{r|}{\\textbf{4.58}}  & \\multicolumn{1}{r|}{0.244}          & 4.601                              & \\multicolumn{1}{r|}{\\textbf{4.57}}  & \\multicolumn{1}{r|}{0.144}          & 4.581                             \n\\end{tabular}\n\\caption{Results of the human subject study showing the number of participants N, average Likert scale value for coherence and sentiment, Krippendorf's alpha showing inter-annotator agreement, and Skew. For each model a total of 50 samples were drawn randomly from the test set and rated by 3 annotators each, resulting in 150 data points per algorithm.}\n\\label{app:totto_agreement}\n\\end{table}\n\n\\begin{table}[]\n\\centering\n\\footnotesize\n\\begin{tabular}{|l|l|rr|rr|}\n\\multicolumn{1}{|c|}{\\multirow{2}{*}{\\textbf{Group 1}}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{Group 2}}} & \\multicolumn{2}{c|}{\\textbf{Coherence}}                                                      & \\multicolumn{2}{c|}{\\textbf{Correctness}}                                                    \\\\\n\\multicolumn{1}{|c|}{}                                  & \\multicolumn{1}{c|}{}                                  & \\multicolumn{1}{c|}{\\textbf{Diff (G2-G1)}} & \\multicolumn{1}{c|}{\\textit{\\textbf{p-values}}} & \\multicolumn{1}{c|}{\\textbf{Diff (G2-G1)}} & \\multicolumn{1}{c|}{\\textit{\\textbf{p-values}}} \\\\ \\hline\nPPO                                                     & NLPO                                                   & \\multicolumn{1}{r|}{\\textbf{-0.507}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.613}}       & \\textbf{0.001}                                  \\\\\nPPO                                                     & NLPO+Supervised                                        & \\multicolumn{1}{r|}{\\textbf{1.827}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.340}}        & \\textbf{0.001}                                  \\\\\nPPO                                                     & Supervised                                             & \\multicolumn{1}{r|}{\\textbf{1.833}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.313}}        & \\textbf{0.001}                                  \\\\\nPPO                                                     & PPO+Supervised                                         & \\multicolumn{1}{r|}{\\textbf{1.813}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.253}}        & \\textbf{0.001}                                  \\\\\nPPO                                                     & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-1.120}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-1.293}}       & \\textbf{0.001}                                  \\\\\nNLPO                                                    & NLPO+Supervised                                        & \\multicolumn{1}{r|}{\\textbf{2.333}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.953}}        & \\textbf{0.001}                                  \\\\\nNLPO                                                    & Supervised                                             & \\multicolumn{1}{r|}{\\textbf{2.340}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.927}}        & \\textbf{0.001}                                  \\\\\nNLPO                                                    & PPO+Supervised                                         & \\multicolumn{1}{r|}{\\textbf{2.320}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.867}}        & \\textbf{0.001}                                  \\\\\nNLPO                                                    & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-0.613}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.680}}       & \\textbf{0.001}                                  \\\\\nNLPO+Supervised                                         & Supervised                                             & \\multicolumn{1}{r|}{0.007}                 & 0.9                                             & \\multicolumn{1}{r|}{\\textbf{-0.027}}       & \\textbf{0.009}                                  \\\\\nNLPO+Supervised                                         & PPO+Supervised                                         & \\multicolumn{1}{r|}{\\textbf{-0.013}}       & \\textbf{0.009}                                  & \\multicolumn{1}{r|}{\\textbf{-0.087}}       & \\textbf{0.009}                                  \\\\\nNLPO+Supervised                                         & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-2.947}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-2.633}}       & \\textbf{0.001}                                  \\\\\nSupervised                                              & PPO+Supervised                                         & \\multicolumn{1}{r|}{\\textbf{-0.020}}       & \\textbf{0.009}                                  & \\multicolumn{1}{r|}{\\textbf{-0.060}}       & \\textbf{0.009}                                  \\\\\nSupervised                                              & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-2.953}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-2.607}}       & \\textbf{0.001}                                  \\\\\nPPO+Supervised                                          & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-2.933}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-2.547}}       & \\textbf{0.001}                                 \n\\end{tabular}\n\\caption{Results of an post-hoc Tukey HSD Test for difference in means between pairs of algorithms (Group 2 - Group 1) and corresponding $p$-values. Individually statistically significant results are bolded and are used to discuss results in the analysis. Overall $p$-values showing that there is a significant difference in means between the models via a one-way ANOVA test are significant with $p \\ll 0.05$ for both coherence and sentiment.}\n\\label{app:totto:human_tukey}\n\\end{table}\n\n\\clearpage\n\n\\subsubsection{Human Participant Study}\n\nFigure~\\ref{fig:description_interface_totto} shows the ToTTo instructions, example, and interface used for the human evaluation experiments. We made small modifications to the original code release's HTML renderer to make the tables display in our HITs.\nTables~\\ref{app:totto_agreement},~\\ref{app:totto:human_tukey} show averaged results, annotator agreement, and the results of statistical significance tests to determine which models output better generations when rated by humans.\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=.45\\linewidth]{figures\/totto_human_study\/totto_human_1.png}\n    \\includegraphics[width=.45\\linewidth]{figures\/totto_human_study\/totto_human2_2.png}\n    \\includegraphics[width=.45\\linewidth]{figures\/totto_human_study\/totto_human2_3.png}\n    \\includegraphics[width=.45\\linewidth]{figures\/totto_human_study\/totto_human3.png}\n    \\caption{Instructions, two examples, and interface for the ToTTo table description task.}\n    \\label{fig:description_interface_totto}\n\\end{figure*}\n\n\\clearpage\n\\subsubsection{Qualitative Analysis}\n\\label{app:totto_qualitative}\nWe show sample generations from each of the algorithms for three randomly picked prompts below.\n\n\\begin{lstlisting}\nSample 1\nPrompt: <table> <cell> August 1, 1992 <col_header> Date <\/col_header> <col_header> North America <\/col_header> <col_header> Australasia <\/col_header> <col_header> Asia <\/col_header> <col_header> North America <\/col_header> <\/cell> <cell> Allentown <col_header> City <\/col_header> <col_header> North America <\/col_header> <col_header> Australasia <\/col_header> <col_header> Asia <\/col_header> <col_header> North America <\/col_header> <\/cell> <cell> Allentown Fairgrounds Grandstand <col_header> Country <\/col_header> <col_header> North America <\/col_header> <col_header> Australasia <\/col_header> <col_header> Asia <\/col_header> <col_header> North America <\/col_header> <\/cell> <\/table><page_title> Under My Spell Tour <\/page_title> <section_title> Tour dates <\/section_title> <table> <cell> August 1, 1992 <col_header> Date <\/col_header> <col_header> North America <\/col_header> <col_header> Australasia <\/col_header> <col_header> Asia <\/col_header> <col_header> North America <\/col_header> <\/cell> <cell> Allentown <col_header> City <\/col_header> <col_header> North America <\/col_header> <col_header> Australasia <\/col_header> <col_header> Asia <\/col_header> <col_header> North America <\/col_header> <\/cell> <cell> Allentown Fairgrounds Grandstand <col_header> Country <\/col_header> <col_header> North America <\/col_header> <col_header> Australasia <\/col_header> <col_header> Asia <\/col_header> <col_header> North America <\/col_header> <\/cell> <\/table>\nZero-Shot: table> cell> Under My Spell Tour \/section_title> Tour dates \/cell> cell> August 1, 1992 col_header> Date \/col_header>\nPPO: Under My Spell Tour, August 1, 1992. August 1, 1992.\nNLPO: Under My Spell Tour dates Under My Spell Tour dates Allentown Fairgrounds Grandstand August 1, 1992. Under My Spell Tour dates Under My Spell Tour dates Allentown Fairgrounds Grandstand August 1, 1992 table>\nSupervised: The Under My Spell Tour began on August 1, 1992 at the Allentown Fairgrounds Grandstand in Allentown.\nSupervised + PPO: On August 1, 1992, Under My Spell Tour was held at the Allentown Fairgrounds Grandstand in Allentown.\nSupervised + NLPO: On August 1, 1992, Under My Spell Tour was held at the Allentown Fairgrounds Grandstand in Allentown.\n\nSample 2\nPrompt: <table> <cell> Philip IV the Fair, the Iron King (Philippe) <col_header> Name <\/col_header> <\/cell> <cell> 5 October 1285 <col_header> King from <\/col_header> <\/cell> <cell> King of France and of Navarre (Roi de France et de Navarre) <col_header> Title <\/col_header> <\/cell> <\/table><page_title> List of French monarchs <\/page_title> <section_title> House of Capet (987\u20131328) <\/section_title> <table> <cell> Philip IV the Fair, the Iron King (Philippe) <col_header> Name <\/col_header> <\/cell> <cell> 5 October 1285 <col_header> King from <\/col_header> <\/cell> <cell> King of France and of Navarre (Roi de France et de Navarre) <col_header> Title <\/col_header> <\/cell> <\/table>\nZero-Shot:\nPPO: Philip IV the Fair, the Iron King (Philippe) 5 October 1285. Title: King of France and of Navarre (Roi de France et de Navarre\nNLPO: King (Philippe) Philip IV the Fair, the Iron King (Philippe) 5 October 1285\nSupervised: Philip IV the Fair, the Iron King (Philippe) was King of France and Navarre from 1285.\nSupervised + PPO: Philip IV the Fair, the Iron King was King of France and Navarre from 1285.\nSupervised + NLPO: Philip IV the Fair, the Iron King (Philippe) was King of France and Navarre (Roi de France et de Navarre) in 1285.\n\nSample 3\nPrompt: <table> <cell> Benjamin A. Smith II (D) <col_header> Vacator <\/col_header> <\/cell> <cell> Successor elected November 6, 1962. <col_header> Reason for change <\/col_header> <\/cell> <cell> Ted Kennedy (D) <col_header> Successor <\/col_header> <\/cell> <cell> November 7, 1962 <col_header> Date of successor's formal installation <\/col_header> <\/cell> <\/table><page_title> 87th United States Congress <\/page_title> <section_title> Senate <\/section_title> <table> <cell> Benjamin A. Smith II (D) <col_header> Vacator <\/col_header> <\/cell> <cell> Successor elected November 6, 1962. <col_header> Reason for change <\/col_header> <\/cell> <cell> Ted Kennedy (D) <col_header> Successor <\/col_header> <\/cell> <cell> November 7, 1962 <col_header> Date of successor's formal installation <\/col_header> <\/cell> <\/table>\nZero-Shot: cell> Ted Kennedy (D) col_header> Vacator \/col_header> \/cell> cell> November 7, 1962 col_header> Reason for change\nPPO: Ted Kennedy (D) succeeding November 6, 1962 as Vacator of the 87th United States Congress.\nNLPO: Ted Kennedy (D) Successor elected November 6, 1962. table> cell Benjamin A. Smith II (D) col_header Reason for change: Date of successor's formal installation November 7, 1962. \nSupervised:  Benjamin A. Smith II (D) served as senate until November 6, 1962 which was later served by Ted Kennedy (D) from November 7, 1962.\nSupervised + PPO: Benjamin A. Smith II (D) served until November 6, 1962 and Ted Kennedy (D) succeeded him from November 7, 1962.\nSupervised + NLPO: Benjamin A. Smith II (D) served until November 6, 1962 and Ted Kennedy (D) succeeded him from November 7, 1962.\n\\end{lstlisting}\n\n\\subsubsection{PPO All-in-One}\n\\kb{Borrowed from: https:\/\/spinningup.openai.com\/en\/latest\/algorithms\/ppo.html }\n\\begin{algorithm}[h]\n    \\caption{PPO}\n    \\begin{algorithmic}\n      \\STATE {\\bfseries Input:} initial policy parameters $\\theta_0$\n      \\STATE {\\bfseries Input:} initial value function parameters $\\phi_0$\n      \\REPEAT\n          \\STATE Collect set of trajectories $\\mathcal{D}_m = \\{  \\tau_i\\}$ by running policy $\\pi_{\\theta_m}$ in the environment.\n          \\STATE Compute rewards-to-go $\\hat{R}_t$\n          \\STATE Compute the advantage estimate $\\hat{A}_t$\n          \\STATE Update the policy by maximizing the PPO-Clip objective:\\\\\n          $$\\theta_{m+1} = \\text{argmax}_{\\theta} \\frac{1}{\\vert \\mathcal{D}_m\\vert T } \\sum_{\\tau \\in \\mathcal{D}} \\sum_{\\tau=0}^{T} \\min \\Big( \\frac{\\pi_{\\theta}(a_t \\vert s_t)}{\\pi_{\\theta_m}(a_t \\vert s_t)} A^{\\pi_{\\theta_m}}, g(\\epsilon, A^{\\pi_{\\theta_m}}(s_t, a_t)) \\Big)$$\\\\\n          \\STATE Fit value function by regression on mean-squared error:\\\\\n          $$\\phi_{m+1}= \\text{argmin}_\\phi \\frac{1}{\\vert \\mathcal{D}_m \\vert T} \\sum_{\\tau \\in \\mathcal{D}_m} \\sum_{t=0}^{T} \\Big( V_\\phi(s_t) - \\hat{R}_t \\Big)^2$$\\\\\n      \\UNTIL{convergence}\n    \\end{algorithmic}\n  \\end{algorithm}\n\n\n\\clearpage\n\\subsubsection{Ablation Study}\nBelow are list of things to studied:\n\\begin{itemize}\n    \\item Effect of decoding during rollouts and how it affects exploration and stability\n        \\subitem(https:\/\/arxiv.org\/pdf\/2202.11818.pdf)\n    \\item  Consistent Dropout for Policy Gradient Reinforcement Learning \n    \\item  NLPO hyperparams (top p, target update frequency)\n    \\item  Reward hacking on PPO (vs NLPO)  - with and without penalty term\n    \n\\end{itemize}\n\n\n\\subsubsection{Hyperparameters}\n\n\n\n\\subsection{Environments: Generation as a Token-level MDP}\n\\label{sec:rl4lm_env}\n\n\n\nEach environment is an NLP task: we are given a supervised dataset $\\dataset=\\{({\\boldsymbol{x}} ^{i}, {\\boldsymbol{y}}^{i})\\}_{i=1}^{N}$ of $N$ examples, where ${\\boldsymbol{x}}  \\in \\sinput$ is an language input and ${\\boldsymbol{y}} \\in \\soutput$ is the target string. %\nGeneration %\ncan be viewed as a Markov Decision Process (MDP) $\\langle \\stateSpace, \\actionSpace,\\rewardfunc, \\transFnDef, \\discount, \\hor\\rangle$ using a finite vocabulary $\\vocab$.\nEach episode in the MDP begins by sampling a datapoint $({\\boldsymbol{x}} , {\\boldsymbol{y}})$ from our dataset and ends when the current time step $t$ exceeds the horizon $\\hor$ or an end of sentence (EOS) token is generated. \nThe input ${\\boldsymbol{x}} =(x_0, \\cdots, x_m)$ is a task-specific prompt that is used as our initial state $\\boldsymbol{s}_0=(x_0, \\cdots, x_m)$, where $\\boldsymbol{s}_0\\in \\stateSpace$ and $\\stateSpace$ is the state space with $x_m \\in \\vocab$.  %\nAn action in the environment $\\action_t \\in \\actionSpace$ consists of a token from our vocabulary $\\vocab$. The transition function  $\\transFnDef: \\stateSpace \\times \\actionSpace \\rightarrow \\stateSpace$ deterministically appends an action $\\action_t$ to the end of the state $\\boldsymbol{s}_{t-1}=(x_0, \\cdots, x_m,\\action_0, \\cdots, \\action_{t-1})$. %\nThis continues until the end of the horizon $t \\leq \\hor$ and we obtain a state $\\boldsymbol{s}_\\hor=(x_0, \\cdots, x_m,\\action_0,\\cdots,\\action_T)$.\nAt the end of an episode a reward $\\rewardfunc: \\stateSpace \\times \\actionSpace \\times \\soutput \\rightarrow \\mathbb{R}^{1}$ that depends on the ($\\boldsymbol{s}_\\hor, {\\boldsymbol{y}}$) (e.g., an automated metric like PARENT \\cite{dhingra2019handling}) is emitted. %\n\\framework{} provides an OpenAI gym~\\citep{brockman2016openai} style \nAPI for an RL environment  \nthat simulates this LM-Based MDP formulation.\nAbstracting the details of the MDP environment structure allows for new tasks to be added quickly with compatibility across all implemented algorithms. %\n\n\n\n\n\n\n\n\\subsection{Reward Functions and Evaluation Metrics}\n\\label{sec:rl4lm_reward}\n\nBecause \\framework{} provides a generic interface for per-token or per-sequence generation rewards, it is possible to quickly apply a wide array of RL algorithms to a similarly diverse range of textual metrics-as-rewards. Specifically, we provide interfaces to 1) \\textbf{n-gram overlap metrics} metrics such as ROUGE \\citep{lin2004rouge}, BLEU \\citep{papineni2002bleu}, SacreBLEU~\\citep{post2018call}, METEOR \\citep{banerjee2005meteor}; (2)  \\textbf{model-based semantic metrics} such as BertScore~\\citep{zhang2019bertscore} and BLEURT~\\citep{sellam-2020-bleurt}\nwhich generally provide higher correlation with human judgment;\n3) \\textbf{task-specific metrics} such as\nCIDER~\\citep{vedantam2015cider}, SPICE~\\citep{anderson2016spice}\n(for captioning\/commonsense generation), PARENT~\\citep{dhingra2019handling} (for data-to-text)\nand SummaCZS~\\citep{laban2022summac} (for factuality of summarization); 4)  \n\\textbf{diversity\/fluency\/naturalness metrics} \n such as perplexity, Mean Segmented Type Token Ratio (MSSTR) \\citep{johnson1944studies}, Shannon entropy over unigrams and bigrams \\citep{shannon1948mathematical}, the ratio of distinct n-grams over the total number of n-grams (Distinct-1, Distinct-2) and count of n-grams that appear only once in the entire generated text \\citep{li2015diversity}.\n\n\n\n\n\\subsection{On-policy Actor-critic Algorithms}\n\\label{sec:rl4lm_onpolicy}\n\\framework{} supports fine-tuning and training LMs from scratch via on-policy actor-critic algorithms on language environments.\nFormally, this class of algorithms allows us to train a parameterized control policy defined as $\\pi_\\theta: \\mathcal{S} \\rightarrow \\mathcal{A}$, a function that attempts to select an action in a given state so as to maximize long term discounted rewards over a trajectory $\\mathbb{E}_{\\pi} [\\sum_{t=0}^{T}\\gamma^t \\mathcal{R}(\\boldsymbol{s}_t, a_t)]$.\nOur benchmark experiments focus on fine-tuning a pre-trained LM denoted as $\\lm$ from which we initial our agent's policy $\\pi_\\theta=\\lm$.\nSimilarly, the value network $V_\\phi$ used to estimate the value function is also initialized from $\\lm$ except for the final layer which is randomly initialized to output a single scalar value.\nAs with other deep RL actor-critic algorithms, we define our value and Q-value functions as $V_t^{\\pi}=\\mathbb{E}_{a_t \\sim \\pi}[\\sum_{\\tau=t}^{\\hor} \\discount R(\\boldsymbol{s}_{\\tau}, a_{\\tau}, {\\boldsymbol{y}})]$, \n$Q_t^{\\pi}(\\boldsymbol{s}_t,a_t)= R(\\boldsymbol{s}_{t}, a_{t},  {\\boldsymbol{y}}) + \\discount \\mathbb{E}_{\\state_{t+1} \\sim \\transFnDef}[V_{t+1}^{\\pi}(\\boldsymbol{s}_{t+1})]$ \nleading to a definition of our advantage function as $A_t^{\\pi}(\\boldsymbol{s},a)=Q_t^{\\pi}(\\boldsymbol{s},a)-V_t^{\\pi}$.\nTo increase training stability, advantage is appoximated using Generalized Advantage Estimation~\\citep{schulman2015high}.\n\nGiven an input-output pair $({\\boldsymbol{x}} , {\\boldsymbol{y}})$ and generation predictions from our agent; because the environment rewards are sequence-level and sparse, following \\cite{wu2021recursively} we regularize the reward function using a token-level KL penalty for all on-policy algorithms, to prevent the model from deviating too far from the initialized LM $\\lm$. Formally, the regularized reward function is:\n\\begin{equation}\n   \\hat{R}(\\boldsymbol{\\state}_t, \\action_t, {\\boldsymbol{y}}) = R(\\boldsymbol{\\state}_t, \\action_t, {\\boldsymbol{y}}) - \\beta \\text{KL}\\left( \\pi_\\theta(\\action_t| \\boldsymbol{\\state}_t) || \\lm(\\action_t| \\boldsymbol{\\state}_t) \\right)\n   \\label{eq:rewardstogo}\n\\end{equation}\nwhere $\\hat{R}$ is the regularized KL reward,  ${\\boldsymbol{y}}$ is gold-truth predictions, $\\text{KL}\\left( \\pi_\\theta(\\action_t| \\boldsymbol{\\state}_t) || \\lm(\\action_t| \\boldsymbol{\\state}_t) \\right) = (\\log \\lm(\\action_t| \\boldsymbol{\\state}_t) - \\log \\pi(\\action_t| \\boldsymbol{\\state}_t))$ and the KL coefficient $\\beta$ is dynamically adapted~\\citep{ziegler2019fine}.\nFurther details on actor-critic methods can be found in Appendix~\\ref{app:onpolicy}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Potential Pitfalls of Reinforcement Learning  \\kb{Something more catchy?}}\nThough reformulating NLP tasks seems enticing, there has been some problems and concerns mention in the literature \\kb{A better catchy phrase}. \n\\textcolor{blue}{(1)} In, past issues, regarding optimizing a reward for language (especially for a metric) does not guarantee an increase in the quality of the output \\citep{paulus2017deep} (i.e. reward hacking \\citep{}). \n\\textcolor{blue}{(5)} In addition, issues around solving a sparse reward problem for text-generation \\citep{Choshen2020On}. (6)\n\\textcolor{blue}{(2)} \\kb{[Raj check this $\\rightarrow$] Also, concerns around RL is only likely to improve over pre-trained models when the pre-trained is already performing well \\citep{Choshen2020On}; including using the KL penalty and staying extremely close to the pre-trained model\\citep{korbak2022controlling} }.\n\\textcolor{blue}{(3)} RL methods have high variance and instability in the objective function \\citep{wu2018study,li-etal-2017-adversarial, wu2016google}. \n\\textcolor{blue}{(4)} The action space in RL for text-generation is high dimensional and combinatorial (size of the vocabulary) \\citep{Choshen2020On}. \n\\textcolor{blue}{(5)} \\kb{[Raj check this $\\rightarrow$] RL methods suffer from loss of capabilities of the original model} \\citep{korbak2022controlling}. \n\\textcolor{blue}{(6)} \\kb{[Raj check this $\\rightarrow$] Sum of actions for rewards not equal to the overall metrics, forcing a discount factor $\\discount \\approx 1$, in most RL for text-generation experiments \\citep{}} \n\\textcolor{blue}{(7)} \\kb{[Raj check this $\\rightarrow$] Extremely high initial state distribution $\\initialstate$ due to prompts needed for encoder-decoders to reduce instability with PPO. } \\citep{}.\nAlthough some of these issues has been addressed only in the context of Neural Machine Translation (NMT) \\citet{samuel2021revisiting}, this evaluation focuses on a broader set of text-generation task.\n\n\n\n\n\n\\subsection{Guidelines on When to Use RL for Text-Generation}\nWhat is RL good for in the context of language? Metric optimization (be careful), sequential tasks, intermediate rewards - incorporating feedback (interactivity), goal driven generation, already have a good init policy via pretraining\n\n\npart of the fix is the KL penalty, but let's think about MDP sample complexity, that depends on gamma too\n\n\n\n\n\\subsection{Tasks and Metrics}\n\n\\begin{table*}[t!]\n  \\small\n  \\begin{center}\n  \\begin{tabular}{@{}m{8.2em}m{11.2em}m{5em}rm{5em}rm{3em}rm{3em}@{}}\n  \\toprule\n       \\textbf{Dataset}\n     & \\textbf{Task}\n     & \\centering\\arraybackslash \\textbf{Language(s)}\n     & \\textbf{Size}\n     & \\centering\\arraybackslash\\textbf{Input Type} \n     & \\centering\\arraybackslash\\textbf{\\textcolor{red}{Done}}\\\\\n  \\toprule\n      \\makecell[bl]{\\small{CommonGEN}\\\\ \\small{\\citep{lin2019commongen}}}\n      & \\small{Generative Commonsense}\n      & \\centering\\arraybackslash\\small{en} \n      & \\centering\\arraybackslash\\small{67k} \n      & \\centering\\arraybackslash\\small{Concept Set} \n      & .\n      & Y\n      \\\\ \n  \\midrule\n      \\makecell[bl]{\\small{CNN Daily Mail}}\n      & \\small{Summarization}\n      & \\centering\\arraybackslash\\small{en} \n      &  \\centering\\arraybackslash\\small{287k}\n      &  \\centering\\arraybackslash\\small{News Article} \n      & .\n      & Y\n      \\\\\n  \\midrule\n      \\makecell[bl]{\\small{IMDB}\\\\ } \n      & \\small{Text Continuation}\n      &  \\centering\\arraybackslash\\small{en}\n      &  \\centering\\arraybackslash\\small{25k}\n      & \\centering\\arraybackslash\\small{Partial Movie Review} \n      & . %\n      & Y\n      \\\\\n  \\midrule\n      \\makecell[bl]{\\small{ToTTo}\\\\ \\small{\\citep{parikh2020totto}}} \n      & \\small{Data to Text}\n      & \\centering\\arraybackslash\\small{en} \n      & \\centering\\arraybackslash\\small{136k}  \n      & \\centering\\arraybackslash\\small{Highlighted Table}\n      & .\n      & Y\n      \\\\\n  \\midrule\n      \\makecell[bl]{\\small{WMT-19}} \n      & \\small{Machine Translation}\n      & \\centering\\arraybackslash\\small{en, ru, de, fr} \n      & \\centering\\arraybackslash\\small{-}  \n      & \\centering\\arraybackslash\\small{Text}\n      & .\n      & Y\n      \\\\\n  \\midrule\n      \\makecell[bl]{\\small{NarrativeQA}}\n      & \\small{Question Answering}\n      & \\centering\\arraybackslash\\small{en} \n      & \\centering\\arraybackslash\\small{-}  \n      & \\centering\\arraybackslash\\small{Question Context}\n      & .\n      & Y\n      \\\\\n  \\bottomrule\n  \\end{tabular}\n  \\end{center}\n  \\caption{}\n  \\label{tab:overview}\n\\end{table*}\n\n\n\n\n\\subsection{On-Policy Algorithms for Generation}\n\nhow do we actually get these to work, what's the key idea - restrict optimization to trust region of \"natural language\"\n\nwell, that's slow - so we use the KL penalty which \n\ngeneric actor critic for generation psuedocode\n\nthe calculate returns line varies based on PPO vs A2C. TRPO straight up does second order\n\n\\subsection{Results on GRUE: Which Algorithm Should be Used to Learn Preferences?}\n\\label{sec:grue_warm_start}\n\nFigures~\\ref{fig:auto_task},~\\ref{fig:auto_natural} present the results on GRUE{}, split into task metrics and naturalness metrics, and Tables~\\ref{tab:key_results},~\\ref{tab:imdb_ablations} highlight key results via ablation studies. Full results are available in Appendix~\\ref{app:experiments}. %\nFor text continuation and summarization, with non-trivial zero-shot performance, RL tends to perform better than supervised training, but for tasks like Commongen and ToTTo, which have very low zero-shot performance, supervised training performs best. \nAs expected, both RL and supervised learning outperform zero-shot approaches.\n\nHowever, \\textbf{using RL+Supervised learning in conjunction works best;}  \nNLPO+supervised and PPO+supervised usually always outperforms NLPO\/PPO (or supervised in isolation) across both task metrics and naturalness metrics. %\nSupervised warm-starting is particularly effective for Commongen and ToTTo, which our results suggest are more prone to reward hacking. \nWe note that Supervised+NLPO using a T5-base (220m parameter) LM currently outperforms all the models on the ToTTo leaderboard, many of which have $\\geq$ 3b parameter supervised models---suggesting that RL is parameter efficient as well.\nIn these cases, it is critical that the initial policy already contain (some) signal for the task: this is because the initial policy is used both as a KL constraint, and as additional masking constraint in NLPO. \nIf the mask contains no initial priors about task specific language, it will be eliminating the wrong actions. Simply put, the better one's initial policy, the more additional performance RL training will add to it.\n\n\n\n\n\\textbf{Human agreement with automated metrics.}\nAs human judgments can be noisy, we run additional statistical analysis such as measuring\ninter-annotator agreement, via Krippendorf's alpha score, and using a one-way ANOVA followed by a post-hoc Tukey HSD test to measure if differences in means of average scores between pairs of models are significant.\nWe find that trends in our human evaluations generally match those seen in the automated metrics for both task and naturalness metrics (see Figures~\\ref{fig:human_task},~\\ref{fig:human_natural} which summarize Appendix Tables~\\ref{app:imdb:human_tukey},\\ref{app:commongen:human_tukey},\\ref{app:summariazation:human_tukey},\\ref{app:totto:human_tukey}---Supervised+NLPO $>$ Supervised $\\geq$ Supervised+PPO $>$ NLPO $\\geq$ PPO $>$ Zero-shot---with the exception of Supervised outperforming Supervised+PPO on 2 out of 4 tasks tasks when automated metrics would indicate that Supervised+PPO outperforms Supervised on all of the tasks.\nWe draw two conclusions from this: (1) if the generated text is above a certain threshold of naturalness, the automated metrics \\textit{usually} correlate with human judgements; (2) usually but not always as seen in the relative performance of Supervised and Supervised+PPO, potentially indicating reward hacking behaviors undetected by automated metrics but caught by human preference feedback.\n\n\n\n\n\\begin{table}\n\\parbox{.6\\linewidth}{\n\\centering\n\\resizebox{0.6\\columnwidth}{!}{\n\\begin{tabular}{lcccccc}\n\\toprule\n{\\cellcolor{gray!25}\\textbf{Questions}}\n& \\multicolumn{6}{c}{\\cellcolor{gray!25}\\textbf{Tasks}} \\\\ \n& IMDB & CommonGen & CNN\/DM & ToTTO & IWSLT-17 & NarQA \\\\\n\\midrule\nNeeds Warm Start & \\cellcolor{green!25}\\ding{55} & \\cellcolor{red!25}\\ding{51} & \\cellcolor{red!25}\\cellcolor{green!25}\\ding{55} & \\cellcolor{red!25}\\ding{51} & \\cellcolor{green!25}\\ding{55} & \\cellcolor{red!25}\\ding{51}\\\\\nEasily reward hackable? & \\cellcolor{red!25}\\ding{51} & \\cellcolor{red!25}\\ding{51} & \\cellcolor{green!25}\\ding{55} & \\cellcolor{green!25}\\ding{55} & \\cellcolor{green!25}\\ding{55} & \\cellcolor{green!25}\\ding{55}\\\\\n\nRL $>$ Sup (auto)? & \\cellcolor{green!25}\\ding{51} & \\cellcolor{red!25}\\ding{55} & \\cellcolor{red!25}\\ding{55} & \\cellcolor{red!25}\\ding{55} & \\cellcolor{red!25}\\ding{55} & \\cellcolor{red!25}\\ding{55} \\\\\nRL $>$ Sup (human)? & \\cellcolor{green!25}\\ding{51} & \\cellcolor{red!25}\\ding{55} & \\cellcolor{red!25}\\ding{55} & \\cellcolor{red!25}\\ding{55} & - & - \\\\\nSup+RL $>$ Sup (auto)? & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51}\\\\\nSup+RL $>$ Sup (human)? & \\cellcolor{green!25}\\ding{51} & \\cellcolor{red!25}\\ding{55} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & - & - \\\\\nSup+NLPO $>$ Sup+PPO (auto)? & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51}\\\\\nSup+NLPO $>$ Sup+PPO (human)? & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & - & -\\\\\n\n\n\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{Key questions answered using GRUE + RL4LMs:} This table summarizes the results found in the ablations and Fig.~\\ref{fig:allmetrics} and provides an overview of the questions we ask in Section~\\ref{sec:benchmark_analysis}: which tasks require warm starts or are easily reward hackable; when to use RL over Supervised, when to use both; and when to use NLPO over PPO. All conclusions drawn are the result of statistical analysis as discussed in the experimental setup.\n}\n\\label{tab:key_results}\n}\n\\parbox{.4\\linewidth}{\n\\centering\n\\resizebox{0.4\\columnwidth}{!}{\n\\begin{tabular}{lrr}\n        \\textbf{Ablation}                    & \\textbf{Sentiment}         & \\textbf{Perplexity}         \\\\ \\hline\n    Zero Shot               & 0.489           & 32.171                    \\\\\n    Supervised              &  0.539  & 35.472                   \\\\\n    PPO                     & 0.605  & 33.497                  \\\\\n    NLPO                    & 0.637            & 32.667                     \\\\ \\hline\n    \\rowcolor{lightgray}\n    \\multicolumn{3}{l}{Warm Starting (Sec.~\\ref{sec:grue_warm_start})}                                \\\\ \\hline\n    PPO+Supervised          & 0.617  &34.078                    \\\\\n    NLPO+Supervised         & 0.645  &33.191                   \\\\ \\hline\n    \\rowcolor{lightgray}\n    \\multicolumn{3}{l}{Data Budget (Reward trained on 10\\% of data, Sec.~\\ref{sec:grue_data_budget})} \\\\ \\hline\n    PPO                     &  0.598   &35.929                   \\\\\n    NLPO                    &  0.599  &33.536                    \\\\ \\hline\n    \\rowcolor{lightgray}\n    \\multicolumn{3}{l}{Removing NLPO Top-$p$ Constraints (Sec.~\\ref{sec:grue_reward})}\\\\\n    \\multicolumn{3}{l}{($p=1$ is equivalent to PPO, $p=0.9$ is NLPO)}                      \\\\ \\hline\n    NLPO $p=0.1$               &  0.579  &32.451                    \\\\\n    NLPO $p=0.5$              &  0.588  &32.447                   \\\\ \\hline\n    \\rowcolor{lightgray}\n    \\multicolumn{3}{l}{Removing KL Constraints (Sec.~\\ref{sec:grue_reward})}                      \\\\ \\hline\n    PPO-no-KL               &  0.859  &37.553                     \\\\\n    NLPO-no-KL              &  0.853  &36.812                   \\\\ \\hline\n    \\rowcolor{lightgray}\n    \\multicolumn{3}{l}{Discount Ablations ($\\gamma=1$) (Sec.~\\ref{sec:grue_practical})}            \\\\ \\hline\n    PPO                     &  0.651  &41.035                     \\\\\n    NLPO                    &  0.624  &43.72                    \n    \\end{tabular}\n    }\n  \\captionof{table}{IMDB Ablation Results.}\n  \\label{tab:imdb_ablations}\n}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\\subsection{Reward Selection and Hacking}\n\\label{sec:grue_reward}\nWhile the GRUE{} benchmark's metric for each task is an average over several measures, the RL models we trained optimized only a single metric independently.\nThus, we can empirically investigate which metric for which GRUE{} produces the best results.\nWe observe that many possible single metric rewards provide task performance gains over supervised methods (results shown in Fig.~\\ref{fig:auto_task_orig},~\\ref{fig:human_task} are averaged across these reward functions) with the condition that the text is also coherent and natural.\n\n\n\\textbf{Which constraints best prevent reward hacking?}\nThe reward function in Equation~\\ref{eq:rewardstogo} balances a task-specific reward with a KL constraint --- models are penalized from straying too far from a base LM in their pursuit of high reward (Table~\\ref{tab:imdb_ablations} and Appendix Table~\\ref{tbl:text_cont_KL_ablations}) clearly show that if KL constraints are removed entirely, models reward hack). %\nBut which model works best as a base regularizing LM?\nWhen the initial policy (i.e., the raw, pretrained model) has low performance on the task, the KL penalty pushes the policy towards nonsense, e.g. on Commongen and ToTTo the trained policy learns to simply repeat portions of the input (as seen in Tables~\\ref{app:tbl_common_gen_qualitative},~\\ref{app:totto_qualitative}).\nThis behavior is mitigated if the base regularizing LM is the supervised model---the reward encourages the policy to balance the task-specific reward and a more reasonable regularization term. \nDeriving KL penalties from warm-started initial policies is critical for performance on such tasks.\n\n\\textbf{PPO vs. NLPO.}\nFigure~\\ref{fig:allmetrics} shows that NLPO generally outperforms PPO and supervised, especially when applied after supervised training.\nWe hypothesize that the primary reason for NLPO's improved performance and stability is because the masking policy provides an additional constraint for the current policy.\nThis constraint is not based on the initial untuned policy like the KL penalty but of the policy from $\\mu$ iterations ago and likely contains more task-relevant information learned during RL training.\nTable~\\ref{tab:imdb_ablations} (and Appendix Table~\\ref{tbl:text_cont_nlpo_hyperparam}) shows how performance increases up to a point and then decreases as $p$ in top-$p$ sampling is increased for the masking policy, relaxing the constraint by eliminating less tokens at each step, implying that there is a balance to be found in how much the model should be constrained during RL training.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Data Budget: Improve your Reward or Gather More Demonstration?}\n\\label{sec:grue_data_budget}\nGiven a fixed data collection budget, is it more efficient to gather feedback to improve a learned reward function or to gather more expert demonstrations?\nWe use the IMDB text continuation task as a case study.\nIn the IMDB task, a model is given a partial movie review as a prompt, and is asked to continue it as positively as possible (even if the prompt was negative).\nThe original dataset consists of movie reviews and sentiment labels of positive, negative, or neutral.\nA DistilBERT~\\citep{sanh2019distilbert} classifier is trained on these labels and used to provide sentiment scores on how positive a given piece of text is, which serves as the task reward. The (simulated) trade-off is between: 1) gathering more sentiment labels (improving the reward); or 2) gathering more positive sentiment reviews (improving supervised training).\n\nWe train a classifier on varying amounts of training data and evaluate on the held out test dataset---finding as expected that more training data improves test accuracy and so results in a higher quality reward.\nWe then use each of these rewards of varying quality during RL training, and evaluate using the same metric as GRUE{} (i.e., a classifier trained with the entire training set).\nAs seen in Table~\\ref{tab:imdb_ablations}, we find that improving the reward quality %\nimproves LM performance as well.\nFurther, we trained a supervised model with at least as many samples used to train each of these reward classifiers.\nWe find that \\textbf{a learned reward function enables greater performance when used as a signal for an RL method than a supervised method trained with 5 times more data.} \nThis implies that improving reward models can be more data efficient than collection expert demonstrations for a task---and that's not accounting for the fact that assigning sentiment labels is likely a simpler task than writing full demonstrations. %\nFurther details on this ablation are found in Appendix Table~\\ref{tbl:text_cont_extra_data}.\n\n\n\n\n\\subsection{Practical Considerations: Which Implementation Details Matter Most?}\n\\label{sec:grue_practical}\n\n\n\\textbf{Generation as a token-level MDP, not a bandit environment.}\nMost recent works that tune LMs using RL do so by calculating a reward for all the tokens in the sentence %\n~\\citep{wu2021recursively,ouyang2022training,lu2022quark}.\nThis setting is equivalent to a bandit feedback environment where the action space is the space of all possible generations for the task~\\citep{sutton2018reinforcement}.\nThis type of environment can be simulated within our RL formulation by setting the discount factor $\\gamma=1$.\nTable~\\ref{tab:imdb_ablations} (and Appendix Table~\\ref{tbl:text_cont_gamma}) shows that this causes instability in training with respect to naturalness in both PPO and NLPO for IMDB.\nOur standard setting is $\\gamma=0.95$ when calculating discounted rewards-to-go in the token-level MDP formulation, which reduces the magnitude of the reward that is applied to tokens selected at the beginning. \nThe sentiment scores are approximately the same between both settings but the naturalness of language in the bandit setting is significantly less ---indicating that discounting rewards via $\\gamma<1$ via a token-level MDP formulation is at least sometimes more effective for language generation.\n\n\\textbf{Dropout and Sampling.}\nWe found two other implementation details to be critical for stability of RL training.\nThe first is dropout, which in its standard form was found to cause instability in policy gradient methods in continuous control settings by~\\citep{hausknecht2022consistent}.\nWe find a similar effect when using dropout when RL training LMs as well, with training loss often diverging for dropout $> 0$ in training.\nThe second important detail, particularly affecting the machine translation task, is sampling methods. %\nWe find that using the same sampling methods during exploration and inference is critical to translating training performance to test performance--else the model exhibits high train rewards but low test metrics.\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:Intro}\nIn recent years, many combinatorial Hopf algebras, whose bases are indexed\nby combinatorial objects, have been intensively studied. For example,\nthe Malvenuto-Reutenauer Hopf algebra~${\\bf FQSym}$ of Free quasi-symmetric\nfunctions~\\cite{MR95,DHT02} has bases indexed by permutations. This\nHopf algebra admits several Hopf subalgebras: The Hopf algebra of Free\nsymmetric functions~${\\bf FSym}$~\\cite{PR95,DHT02}, whose bases are indexed\nby standard Young tableaux, the Hopf algebra~${\\bf Bell}$~\\cite{R07} whose bases\nare indexed by set partitions, the Loday-Ronco Hopf algebra~${\\bf PBT}$~\\cite{LR98,HNT05}\nwhose bases are indexed by planar binary trees, and the Hopf algebra~${\\bf Sym}$\nof non-commutative symmetric functions~\\cite{GKDLLRT94} whose bases are\nindexed by integer compositions. A unifying approach to construct all these\nstructures relies on a definition of a congruence on words leading to the\ndefinition of monoids on combinatorial objects. Indeed,~${\\bf FSym}$ is directly\nobtained from the plactic monoid~\\cite{LS81,DHT02,Lot02},~${\\bf Bell}$ from the\nBell monoid~\\cite{R07},~${\\bf PBT}$ from the sylvester monoid~\\cite{HNT02,HNT05},\nand~${\\bf Sym}$ from the hypoplactic monoid~\\cite{KT97,N98}. The richness of\nthese constructions relies on the fact that, in addition to constructing\nHopf algebras, the definition of such monoids often brings partial orders,\ncombinatorial algorithms and Robinson-Schensted-like algorithms, of independent\ninterest.\n\\smallskip\n\nThe Baxter combinatorial family admits various representations. The most\nfamous of these are Baxter permutations~\\cite{Bax64}, which are permutations\nthat avoid certain patterns, and pairs of twin binary trees~\\cite{DG94}.\nThis family also contains more exotic objects like quadrangulations~\\cite{ABP04}\nand plane bipolar orientations~\\cite{BBF10}. In this paper, we propose to\nenrich the above collection of Hopf algebras by providing a plactic-like\nmonoid, namely the Baxter monoid, leading to the construction of a Hopf\nalgebra whose bases are indexed by objects belonging to this combinatorial\nfamily.\n\\smallskip\n\nIn order to show examples of relations between lattice congruences~\\cite{CS98}\nand Hopf algebras, Reading presented in~\\cite{Rea05} a lattice congruence of\nthe permutohedron whose equivalence classes are indexed by twisted Baxter\npermutations. These permutations were defined by a pattern avoidance property.\nThis congruence is very natural: The meet of two lattice congruences of\nthe permutohedron related to the construction of~${\\bf PBT}$ is one starting\npoint to build~${\\bf Sym}$; A natural question is to understand what happens\nwhen the join, instead of the meet, of these two lattice congruences is\nconsidered. Reading proved that his lattice congruence is precisely this\nlast one, and that the minimal elements of its equivalence classes are\ntwisted Baxter permutations. Besides, thanks to his theory, he gets for\nfree a Hopf algebra whose bases are indexed by twisted Baxter permutations.\nActually, twisted Baxter permutations are equinumerous with Baxter permutations.\nIndeed, Law and Reading pointed out in~\\cite{LR10} that the first proof\noccurred in unpublished notes of West. Hence, the Hopf algebra of Reading\ndefined in~\\cite{Rea05} can already be seen as a Hopf algebra on Baxter\npermutations, and our construction, considered as a different construction\nof the same Hopf algebra. Moreover, very recently, Law and Reading~\\cite{LR10}\ndetailed their construction of this Hopf algebra and studied some of its\nalgebraic properties.\n\\smallskip\n\nWe started independently the study of Baxter objects in a different way:\nWe looked for a quotient of the free monoid analogous to the plactic and\nthe sylvester monoid. Surprisingly, the equivalence classes of permutations\nunder our monoid congruence are the same as the equivalence classes of the\nlattice congruence of Law and Reading, and hence have the same by-products, as\n\\emph{e.g.}, the Hopf algebra structure and the fact that each class contains\nboth one twisted and one non-twisted Baxter permutation. However, even if\nboth points of view lead to the same general theory, their paths are different\nand provide different ways of understanding the construction, one centered\non lattice theory, the other centered on combinatorics on words. Moreover,\na large part of the results of each paper do not appear in the other as,\nin our case, the Robinson-Schensted-like correspondence and its insertion\nalgorithm, the polynomial realization, the bidendriform bialgebra structure,\nthe freeness, cofreeness, self-duality, primitive elements, and multiplicative\nbases of the Hopf algebra, and a few other combinatorial properties.\n\\smallskip\n\nWe begin by recalling in Section~\\ref{sec:Prelim} the preliminary notions\nabout words, permutations, and pairs of twin binary trees used thereafter.\nIn Section~\\ref{sec:MonoideBaxter}, we define the Baxter congruence. This\ncongruence allows to define a quotient of the free monoid, the Baxter monoid,\nwhich has a number of properties required for the Hopf algebraic construction\nwhich follows. We show that the Baxter monoid is intimately linked to the\nsylvester monoid and that the equivalence classes of the permutations under\nthe Baxter congruence form intervals of the permutohedron. Next, in\nSection~\\ref{sec:RobinsonSchensted}, we develop a Robinson-Schensted-like\ninsertion algorithm that allows to decide if two words are equivalent according\nto the Baxter congruence. Given a word, this algorithm computes iteratively\na pair of twin binary trees inserting one by one the letters of~$u$. We\ngive as well some algorithms to read the minimal, the maximal and the Baxter\npermutation of a Baxter equivalence class encoded by a pair of twin binary\ntrees. We also show that each equivalence class of permutations under the\nBaxter congruence contains exactly one Baxter permutation.\nSection~\\ref{sec:TreillisBaxter} is devoted to the study of some properties\nof the equivalence classes of permutations under the Baxter congruence.\nThis leads to the definition of a lattice structure on pairs of twin binary\ntrees, very similar to the Tamari lattice~\\cite{Tam62,Knu06} since covering\nrelations can be expressed by binary tree rotations. We introduce in this\nsection \\emph{twin Tamari diagrams} that are objects in bijection with pairs\nof twin binary trees and offer a simple way to test comparisons in this\nlattice. Finally, in\nSection~\\ref{sec:AlgebreHopfBaxter}, we start by recalling some basic facts\nabout the Hopf algebra of Free quasi-symmetric functions~${\\bf FQSym}$, and\nthen give our construction of the Hopf algebra~${\\bf Baxter}$ and study it.\nUsing the polynomial realization of~${\\bf FQSym}$, we deduce a polynomial\nrealization of~${\\bf Baxter}$. Using the order structure on pairs of twin\nbinary trees defined in the above section, we describe its product as an\ninterval of this order. Moreover, we prove that this Hopf algebra is free\nas an algebra by constructing two multiplicative bases, and introduce two\noperators on pairs of twin binary trees, analogous to the operators \\emph{over}\nand \\emph{under} of Loday-Ronco on binary trees~\\cite{LR02}. Using the results\nof Foissy on bidendriform bialgebras~\\cite{Foi07}, we show that this Hopf algebra\nis also self-dual and that the Lie algebra of its primitive elements is free.\nWe conclude by explaining some morphism with other known Hopf subalgebras\nof~${\\bf FQSym}$.\n\\medskip\n\nThis paper is an extended version of~\\cite{Gir11}. It contains all proofs\nand Sections~\\ref{sec:RobinsonSchensted} and~\\ref{sec:AlgebreHopfBaxter}\nhave new results.\n\n\\subsubsection*{Acknowledgments}\nThe author would like to thank Florent Hivert and Jean-Christophe Novelli\nfor their advice and help during all stages of the preparation of this paper.\nThe computations of this work have been done with the open-source mathematical\nsoftware Sage~\\cite{SAGE}.\n\n\\section{Preliminaries} \\label{sec:Prelim}\n\n\\subsection{Words, definitions and notations}\nIn the sequel, $A := \\{a_1 < a_2 < \\cdots\\}$ is a totally ordered infinite\nalphabet and~$A^*$ is the free monoid generated by~$A$. Let~$u \\in A^*$.\nWe shall denote by~$|u|$ the length of~$u$ and by~$\\epsilon$ the word of\nlength~$0$. The largest (resp. smallest) letter of~$u$ is denoted by~$\\max(u)$\n(resp.~$\\min(u)$). The \\emph{evaluation}~$\\operatorname{ev}(u)$ of the word~$u$ is the\nnon-negative integer vector such that its $i$-th entry is the number of\noccurrences of the letter~$a_i$ in~$u$. It is convenient to denote by\n$\\operatorname{Alph}(u) := \\left\\{u_i : 1 \\leq i \\leq |u|\\right\\}$ the smallest alphabet\non which~$u$ is defined. We say that~$(i, j)$ is an \\emph{inversion} of~$u$\nif~$i < j$ and~$u_i > u_j$. Additionally,~$i$ is \\emph{descent} of~$u$\nif~$(i, i + 1)$ is an inversion of~$u$.\n\\medskip\n\nLet us now recall some classical operations on words. We shall denote by\n$u^\\sim := u_{|u|} \\dots u_1$ the \\emph{mirror image} of~$u$ and by~$u_{|S}$\nthe \\emph{restriction} of~$u$ on the alphabet~$S \\subseteq A$, that is the\nlongest subword of~$u$ such that $\\operatorname{Alph}(u) \\subseteq S$. Let~$v \\in A^*$.\nThe \\emph{shuffle product}~$\\shuffle$ is recursively defined on the linear\nspan of words~$\\mathbb{Z} \\langle A \\rangle$ by\n\\begin{equation}\n    u \\shuffle v :=\n    \\begin{cases}\n        u & \\mbox{if $v = \\epsilon$,} \\\\\n        v & \\mbox{if $u = \\epsilon$,} \\\\\n        {\\tt a} (u' \\shuffle {\\tt b} v') + {\\tt b} ({\\tt a} u' \\shuffle v')\n        & \\mbox{otherwise, where $u = {\\tt a} u'$, $v = {\\tt b} v'$, and ${\\tt a}, {\\tt b} \\in A$.}\n    \\end{cases}\n\\end{equation}\nFor example,\n\\begin{align} \\begin{split}\n    {\\bf a_1 a_2} \\shuffle a_2 a_1\n        & = {\\bf a_1 a_2} a_2 a_1 + {\\bf a_1} a_2 {\\bf a_2} a_1 +\n            {\\bf a_1} a_2 a_1 {\\bf a_2} + a_2 {\\bf a_1 a_2} a_1 +\n            a_2 {\\bf a_1} a_1 {\\bf a_2} + a_2 a_1 {\\bf a_1 a_2}, \\\\\n        & = a_1 a_2 a_1 a_2 + 2\\, a_1 a_2 a_2 a_1 + 2\\, a_2 a_1 a_1 a_2 +\n            a_2 a_1 a_2 a_1.\n\\end{split} \\end{align}\nLet $A^\\# := \\{a_1^\\# > a_2^\\# > \\cdots\\}$ be the alphabet~$A$ on which\nthe order relation has been reversed. The \\emph{Sch\u00fctzenberger transformation}~$\\#$\nis defined on words by\n\\begin{equation}\n    u^\\# = \\left(u_1 u_2 \\dots u_{|u|} \\right)^\\# :=\n        u_{|u|}^\\# \\dots u_2^\\# u_1^\\#.\n\\end{equation}\nFor example, $(a_5 a_3 a_1 a_1 a_5 a_2)^\\# = a_2^\\# a_5^\\# a_1^\\# a_1^\\# a_3^\\# a_5^\\#$.\nNote that by setting ${A^\\#}^\\# := A$, the transformation~$\\#$ becomes an\ninvolution on words.\n\n\\subsection{Permutations, definitions and notations}\nDenote by~$\\mathfrak{S}_n$ the set of permutations of size~$n$ and by~$\\mathfrak{S}$\nthe set of all permutations. One can see a permutation of size~$n$ as a\nword without repetition of length~$n$ on the first letters of~$A$. We shall\ncall~$i$ a \\emph{recoil} of~$\\sigma \\in \\mathfrak{S}_n$ if~$(i, i + 1)$ is\nan inversion of~$\\sigma^{-1}$. By convention,~$n$ also is a recoil of~$\\sigma$.\n\\medskip\n\nThe \\emph{(right) permutohedron order} is the partial order~${\\:\\leq_{\\operatorname{P}}\\:}$\ndefined on~$\\mathfrak{S}_n$ where~$\\sigma$ is covered by~$\\nu$ if\n$\\sigma = u \\, {\\tt a} {\\tt b} \\, v$ and $\\nu = u \\, {\\tt b} {\\tt a} \\, v$ where\n${\\tt a} < {\\tt b} \\in A$, and~$u$ and~$v$ are words. Recall that one has\n$\\sigma {\\:\\leq_{\\operatorname{P}}\\:} \\nu$ if and only if any inversion of~$\\sigma^{-1}$ also\nis an inversion of~$\\nu^{-1}$.\n\\medskip\n\nLet $\\sigma, \\nu \\in \\mathfrak{S}$. The permutation $\\sigma {\\,\\diagup\\,} \\nu$ is obtained\nby concatenating~$\\sigma$ and the letters of~$\\nu$ incremented by~$|\\sigma|$;\nIn the same way, the permutation $\\sigma {\\,\\diagdown\\,} \\nu$ is obtained by\nconcatenating the letters of~$\\nu$ incremented by~$|\\sigma|$ and~$\\sigma$.\nFor example,\n\\begin{equation}\n    {\\bf 312} {\\,\\diagup\\,} 2314 = {\\bf 312} 5647\n    \\quad \\mbox{and} \\quad\n    {\\bf 312} {\\,\\diagdown\\,} 2314 = 5647 {\\bf 312}.\n\\end{equation}\nA permutation~$\\sigma$ is \\emph{connected} if $\\sigma = \\nu {\\,\\diagup\\,} \\pi$\nimplies $\\nu = \\sigma$ or $\\pi = \\sigma$. Similarly,~$\\sigma$ is\n\\emph{anti-connected} if $\\sigma^\\sim$ is connected. The\n\\emph{shifted shuffle product}~$\\cshuffle$ of two permutations is defined by\n\\begin{equation}\n    \\sigma \\cshuffle \\nu :=\n    \\sigma \\shuffle \\left(\\nu_1\\! +\\! |\\sigma| \\dots \\nu_{|\\nu|}\\! +\\! |\\sigma|\\right).\n\\end{equation}\nFor example,\n\\begin{equation}\n    {\\bf 12} \\cshuffle 21 = {\\bf 12} \\shuffle 43 = {\\bf 12} 43 + {\\bf 1} 4 {\\bf 2} 3 +\n    {\\bf 1} 43 {\\bf 2} + 4 {\\bf 12} 3 + 4 {\\bf 1} 3 {\\bf 2} + 43 {\\bf 12}.\n\\end{equation}\nThe \\emph{standardized word}~$\\operatorname{std}(u)$ of~$u \\in A^*$ is the unique permutation\nof size~$|u|$ having the same inversions as~$u$. For example,\n$\\operatorname{std}(a_3 a_1 a_4 a_2 a_5 a_7 a_4 a_2 a_3) = 416289735$.\n\n\\subsection{Binary trees, definitions and notations}\nWe call \\emph{binary tree} any complete rooted planar binary tree. Recall\nthat a binary tree~$T$ is either a \\emph{leaf} (also called \\emph{empty tree})\ndenoted by~$\\perp$, or a node that is attached through two edges to\ntwo binary trees, called respectively the \\emph{left subtree} and the\n\\emph{right subtree} of~$T$. Let~$\\mathcal{B}\\mathcal{T}_n$ be the set of binary trees\nwith~$n$ nodes and~$\\mathcal{B}\\mathcal{T}$ be the set of all binary trees. We use in the\nsequel the standard terminology (\\emph{i.e.}, \\emph{child}, \\emph{ancestor},\n\\emph{path}, \\emph{etc.}) about binary trees~\\cite{AU94}. In our graphical\nrepresentations, nodes are represented by circles\n\\scalebox{.3}{\n\\begin{tikzpicture}\n    \\node[Noeud](0,0){};\n\\end{tikzpicture}},\nleaves by squares\n\\scalebox{.5}{\n\\begin{tikzpicture}\n    \\node[Feuille](0,0){};\n\\end{tikzpicture}},\nedges by segments\n\\scalebox{.3}{\n\\begin{tikzpicture}\n    \\draw[Arete](0,0)--(-1,-.8);\n\\end{tikzpicture}}\nor\n\\scalebox{.3}{\n\\begin{tikzpicture}\n    \\draw[Arete](0,0)--(1,-.8);\n\\end{tikzpicture}},\nand arbitrary subtrees by big squares like\n\\scalebox{.18}{\\raisebox{.3em}{\n\\begin{tikzpicture}\n    \\node[SArbre]{};\n\\end{tikzpicture}}}.\n\n\\subsubsection{The Tamari order}\nThe \\emph{Tamari order}~\\cite{Tam62,Knu06} is the partial order~${\\:\\leq_{\\operatorname{T}}\\:}$\ndefined on~$\\mathcal{B}\\mathcal{T}_n$ where $T_0 \\in \\mathcal{B}\\mathcal{T}_n$ is covered by $T_1 \\in \\mathcal{B}\\mathcal{T}_n$\nif it is possible to obtain~$T_1$ by performing a \\emph{right rotation}\ninto~$T_0$ (see Figure~\\ref{fig:Rotation}).\n\\begin{figure}[ht]\n    \\centering\n    \\scalebox{.3}{%\n    \\begin{tikzpicture}\n        \\node[Noeud, EtiqClair] (racine) at (0, 0) {};\n        \\node (g) at (-3, -1) {};\n        \\node (d) at (3, -1) {};\n        \\node[Noeud, EtiqFonce] (r) at (0, -2) {\\Huge $y$};\n        \\node[Noeud, EtiqClair] (q) at (-2, -4) {\\Huge $x$};\n        \\node[SArbre] (A) at (-4, -6) {$A$};\n        \\node[SArbre] (B) at (0, -6) {$B$};\n        \\node[SArbre] (C) at (2, -4) {$C$};\n        \\draw[Arete] (racine) -- (g);\n        \\draw[Arete] (racine) -- (d);\n        \\draw[Arete, decorate, decoration = zigzag] (racine) -- (r);\n        \\draw[Arete] (r) -- (q);\n        \\draw[Arete] (r) -- (C);\n        \\draw[Arete] (q) -- (A);\n        \\draw[Arete] (q) -- (B);\n        \\node at (-4, -3) {\\scalebox{2.2}{$T_0 = $}};\n        \\path (4, -2) edge[line width=3pt, ->] node[anchor=south,above,font=\\Huge,Noir!100]{right} (6, -2);\n        \\path (6, -4) edge[line width=3pt, ->] node[anchor=south,above,font=\\Huge,Noir!100]{left} (4, -4);\n        \\node[Noeud, EtiqClair] (racine') at (10, 0) {};\n        \\node (g') at (7, -1) {};\n        \\node (d') at (13, -1) {};\n        \\node[Noeud, EtiqFonce] (r') at (12, -4) {\\Huge $y$};\n        \\node[Noeud, EtiqClair] (q') at (10, -2) {\\Huge $x$};\n        \\node[SArbre] (A') at (8, -4) {$A$};\n        \\node[SArbre] (B') at (10, -6) {$B$};\n        \\node[SArbre] (C') at (14, -6) {$C$};\n        \\draw[Arete] (racine') -- (g');\n        \\draw[Arete] (racine') -- (d');\n        \\draw[Arete, decorate, decoration = zigzag] (racine') -- (q');\n        \\draw[Arete] (q') -- (r');\n        \\draw[Arete] (q') -- (A');\n        \\draw[Arete] (r') -- (B');\n        \\draw[Arete] (r') -- (C');\n        \\node at (14, -3) {\\scalebox{2.2}{$ = T_1$}};\n    \\end{tikzpicture}}\n    \\caption{The right rotation of root $y$.}\n    \\label{fig:Rotation}\n\\end{figure}\nOne has $T_0 {\\:\\leq_{\\operatorname{T}}\\:} T_1$ if and only if starting from~$T_0$, it is possible\nto obtain~$T_1$ by performing some right rotations.\n\n\\subsubsection{Operations on binary trees}\nIf~$L$ and~$R$ are binary trees, denote by $L \\wedge R$ the binary tree\nwhich has~$L$ as left subtree and~$R$ as right subtree. Similarly, if~$L$\nand~$R$ are $A$-labeled binary trees, denote by $L \\wedge_{\\tt a} R$\nthe $A$-labeled binary tree which has $L$ as left subtree,~$R$ as right\nsubtree and a root labeled by~${\\tt a} \\in A$.\n\\medskip\n\nLet $T_0, T_1 \\in \\mathcal{B}\\mathcal{T}$. The binary tree $T_0 {\\,\\diagup\\,} T_1$ is obtained by\ngrafting~$T_0$ from its root on the leftmost leaf of~$T_1$; In the same\nway, the binary tree $T_0 {\\,\\diagdown\\,} T_1$ is obtained by grafting~$T_1$ from\nits root on the rightmost leaf of~$T_0$.\n\\medskip\n\nFor example, for\n\\begin{equation}\n    \\begin{split} T_0 := \\: \\end{split}\n    \\begin{split}\n    \\scalebox{.15}{%\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\node[Feuille](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-2){};\n        \\node[Feuille](4)at(4,-3){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud](5)at(5,0){};\n        \\node[Feuille](6)at(6,-2){};\n        \\node[Noeud](7)at(7,-1){};\n        \\node[Feuille](8)at(8,-2){};\n        \\draw[Arete](7)--(6);\n        \\draw[Arete](7)--(8);\n        \\draw[Arete](5)--(1);\n        \\draw[Arete](5)--(7);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\qquad \\mbox{and} \\qquad\n    \\begin{split} T_1 := \\: \\end{split}\n    \\begin{split}\n    \\scalebox{.15}{%\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0,-1){};\n        \\node[Noeud,Marque1](1)at(1,0){};\n        \\node[Feuille](2)at(2,-3){};\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\node[Feuille](4)at(4,-3){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](3)--(4);\n        \\node[Noeud,Marque1](5)at(5,-1){};\n        \\node[Feuille](6)at(6,-2){};\n        \\draw[Arete](5)--(3);\n        \\draw[Arete](5)--(6);\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(5);\n    \\end{tikzpicture}}\\,\n    \\end{split},\n\\end{equation}\nwe have\n\\begin{equation}\n    \\begin{split}T_0 \\wedge T_1 = \\: \\end{split}\n    \\begin{split}\n    \\scalebox{.15}{%\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Feuille](2)at(2,-4){};\n        \\node[Noeud](3)at(3,-3){};\n        \\node[Feuille](4)at(4,-4){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud](5)at(5,-1){};\n        \\node[Feuille](6)at(6,-3){};\n        \\node[Noeud](7)at(7,-2){};\n        \\node[Feuille](8)at(8,-3){};\n        \\draw[Arete](7)--(6);\n        \\draw[Arete](7)--(8);\n        \\draw[Arete](5)--(1);\n        \\draw[Arete](5)--(7);\n        \\node[Noeud,EtiqClair](9)at(9,0){};\n        \\node[Feuille](10)at(10,-2){};\n        \\node[Noeud,Marque1](11)at(11,-1){};\n        \\node[Feuille](12)at(12,-4){};\n        \\node[Noeud,Marque1](13)at(13,-3){};\n        \\node[Feuille](14)at(14,-4){};\n        \\draw[Arete](13)--(12);\n        \\draw[Arete](13)--(14);\n        \\node[Noeud,Marque1](15)at(15,-2){};\n        \\node[Feuille](16)at(16,-3){};\n        \\draw[Arete](15)--(13);\n        \\draw[Arete](15)--(16);\n        \\draw[Arete](11)--(10);\n        \\draw[Arete](11)--(15);\n        \\draw[Arete](9)--(5);\n        \\draw[Arete](9)--(11);\n    \\end{tikzpicture}}\\,\n    \\end{split},\n\\end{equation}\n\\begin{equation}\n    \\begin{split}T_0 {\\,\\diagup\\,} T_1 = \\: \\end{split}\n    \\begin{split}\n    \\scalebox{.15}{%\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Feuille](2)at(2,-4){};\n        \\node[Noeud](3)at(3,-3){};\n        \\node[Feuille](4)at(4,-4){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud](5)at(5,-1){};\n        \\node[Feuille](6)at(6,-3){};\n        \\node[Noeud](7)at(7,-2){};\n        \\node[Feuille](8)at(8,-3){};\n        \\draw[Arete](7)--(6);\n        \\draw[Arete](7)--(8);\n        \\draw[Arete](5)--(1);\n        \\draw[Arete](5)--(7);\n        \\node[Noeud,Marque1](9)at(9,0){};\n        \\node[Feuille](10)at(10,-3){};\n        \\node[Noeud,Marque1](11)at(11,-2){};\n        \\node[Feuille](12)at(12,-3){};\n        \\draw[Arete](11)--(10);\n        \\draw[Arete](11)--(12);\n        \\node[Noeud,Marque1](13)at(13,-1){};\n        \\node[Feuille](14)at(14,-2){};\n        \\draw[Arete](13)--(11);\n        \\draw[Arete](13)--(14);\n        \\draw[Arete](9)--(5);\n        \\draw[Arete](9)--(13);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\qquad \\mbox{and} \\qquad\n    \\begin{split}T_0 {\\,\\diagdown\\,} T_1 = \\: \\end{split}\n    \\begin{split}\n    \\scalebox{.15}{%\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\node[Feuille](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-2){};\n        \\node[Feuille](4)at(4,-3){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud](5)at(5,0){};\n        \\node[Feuille](6)at(6,-2){};\n        \\node[Noeud](7)at(7,-1){};\n        \\node[Feuille](8)at(8,-3){};\n        \\node[Noeud,Marque1](9)at(9,-2){};\n        \\node[Feuille](10)at(10,-5){};\n        \\node[Noeud,Marque1](11)at(11,-4){};\n        \\node[Feuille](12)at(12,-5){};\n        \\draw[Arete](11)--(10);\n        \\draw[Arete](11)--(12);\n        \\node[Noeud,Marque1](13)at(13,-3){};\n        \\node[Feuille](14)at(14,-4){};\n        \\draw[Arete](13)--(11);\n        \\draw[Arete](13)--(14);\n        \\draw[Arete](9)--(8);\n        \\draw[Arete](9)--(13);\n        \\draw[Arete](7)--(6);\n        \\draw[Arete](7)--(9);\n        \\draw[Arete](5)--(1);\n        \\draw[Arete](5)--(7);\n    \\end{tikzpicture}}\\,\n    \\end{split}.\n\\end{equation}\n\n\\subsubsection{Binary search trees, increasing, and decreasing binary trees}\nAn $A$-labeled binary tree~$T$ is a \\emph{right} (resp. \\emph{left})\n\\emph{binary search tree} if for any node~$x$ labeled by~${\\tt b}$, each\nlabel~${\\tt a}$ of a node in the left subtree of~$x$ and each label~${\\tt c}$ of\na node in the right subtree of~$x$, the inequality ${\\tt a} \\leq {\\tt b} < {\\tt c}$\n(resp. ${\\tt a} < {\\tt b} \\leq {\\tt c}$) holds.\n\\medskip\n\nA binary tree $T \\in \\mathcal{B}\\mathcal{T}_n$ is an \\emph{increasing} (resp. \\emph{decreasing})\n\\emph{binary tree} if it is bijectively labeled on $\\{1, \\dots, n\\}$ and,\nfor any node~$x$ of~$T$, if~$y$ is a child of~$x$, then the label of~$y$\nis greater (resp. smaller) than the label of~$x$.\n\\medskip\n\nThe \\emph{shape}~$\\operatorname{sh}(T)$ of an $A$-labeled binary tree~$T$ is the\nunlabeled binary tree obtained by forgetting its labels.\n\n\\subsubsection{Inorder traversal}\nThe \\emph{inorder traversal} of a binary tree~$T$ consists in recursively\nvisiting its left subtree, then its root, and finally its right subtree\n(see Figure~\\ref{fig:ExempleLectureInfixe}).\n\\begin{figure}[ht]\n    \\centering\n    \\scalebox{.2}{\\begin{tikzpicture}\n        \\node[Feuille](0)at(0.0,-2){};\n        \\node[Noeud,label=below:\\scalebox{3.5}{$a$}](1)at(1.0,-1){};\n        \\node[Feuille](2)at(2.0,-4){};\n        \\node[Noeud,label=below:\\scalebox{3.5}{$b$}](3)at(3.0,-3){};\n        \\node[Feuille](4)at(4.0,-4){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](3)--(4);\n        \\node[Noeud,label=below:\\scalebox{3.5}{$c$}](5)at(5.0,-2){};\n        \\node[Feuille](6)at(6.0,-3){};\n        \\draw[Arete](5)--(3);\n        \\draw[Arete](5)--(6);\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(5);\n        \\node[Noeud,label=below:\\scalebox{3.5}{$d$}](7)at(7.0,0){};\n        \\node[Feuille](8)at(8.0,-4){};\n        \\node[Noeud,label=below:\\scalebox{3.5}{$e$}](9)at(9.0,-3){};\n        \\node[Feuille](10)at(10.0,-4){};\n        \\draw[Arete](9)--(8);\n        \\draw[Arete](9)--(10);\n        \\node[Noeud,label=below:\\scalebox{3.5}{$f$}](11)at(11.0,-2){};\n        \\node[Feuille](12)at(12.0,-3){};\n        \\draw[Arete](11)--(9);\n        \\draw[Arete](11)--(12);\n        \\node[Noeud,label=below:\\scalebox{3.5}{$g$}](13)at(13.0,-1){};\n        \\node[Feuille](14)at(14.0,-3){};\n        \\node[Noeud,label=below:\\scalebox{3.5}{$h$}](15)at(15.0,-2){};\n        \\node[Feuille](16)at(16.0,-3){};\n        \\draw[Arete](15)--(14);\n        \\draw[Arete](15)--(16);\n        \\draw[Arete](13)--(11);\n        \\draw[Arete](13)--(15);\n        \\draw[Arete](7)--(1);\n        \\draw[Arete](7)--(13);\n    \\end{tikzpicture}}\n    \\caption{The sequence $(a, b, c, d, e, f, g, h)$ is the sequence of all\n    nodes of this binary tree visited by the inorder traversal.}\n    \\label{fig:ExempleLectureInfixe}\n\\end{figure}\nWe shall say that a node~$x$ is \\emph{the $i$-th node of}~$T$ if~$x$ is\nthe $i$-th visited node by the inorder traversal of~$T$. In the same way,\na leaf~$y$ is \\emph{the $j$-th leaf of}~$T$ if~$y$ is the $j$-th visited\nleaf by the inorder traversal of~$T$. We also say that~$i$ is the \\emph{index}\nof~$x$ and~$j$ is the \\emph{index} of~$y$. If~$T$ is labeled, its\n\\emph{inorder reading} is the word $u_1 \\dots u_{|u|}$ such that for any\n$1 \\leq i \\leq |u|$, $u_i$ is the label of the $i$-th node of~$T$. Note\nthat when~$T$ is a right (or left) binary search tree, its inorder reading\nis a nondecreasing word.\n\n\\subsubsection{The canopy of binary trees}\nThe \\emph{canopy} (see~\\cite{LR98} and~\\cite{V04})~$\\operatorname{cnp}(T)$ of a binary\ntree~$T$ is the word on the alphabet $\\{0, 1\\}$ obtained by browsing the\nleaves of~$T$ from left to right except the first and the last one, writing~$0$\nif the considered leaf is oriented to the right,~$1$ otherwise (see\nFigure~\\ref{fig:ExempleCanopee}). Note that the orientation of the leaves\nin a binary tree is determined only by its nodes so that we can omit to\ndraw the leaves in our graphical representations.\n\\begin{figure}[ht]\n    \\centering\n    \\scalebox{.20}{%\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Feuille](2)at(2,-3){};\n        \\node[] (2') [below of = 2] {\\scalebox{3}{$0$}};\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,-1){};\n        \\node[Feuille](4)at(4,-4){};\n        \\node[] (4') [below of = 4] {\\scalebox{3}{$1$}};\n        \\node[Noeud](5)at(5,-3){};\n        \\node[Feuille](6)at(6,-4){};\n        \\node[] (6') [below of = 6] {\\scalebox{3}{$0$}};\n        \\draw[Arete](5)--(4);\n        \\draw[Arete](5)--(6);\n        \\node[Noeud](7)at(7,-2){};\n        \\node[Feuille](8)at(8,-3){};\n        \\node[] (8') [below of = 8] {\\scalebox{3}{$0$}};\n        \\draw[Arete](7)--(5);\n        \\draw[Arete](7)--(8);\n        \\draw[Arete](3)--(1);\n        \\draw[Arete](3)--(7);\n        \\node[Noeud](9)at(9,0){};\n        \\node[Feuille](10)at(10,-3){};\n        \\node[] (10') [below of = 10] {\\scalebox{3}{$1$}};\n        \\node[Noeud](11)at(11,-2){};\n        \\node[Feuille](12)at(12,-3){};\n        \\node[] (12') [below of = 12] {\\scalebox{3}{$0$}};\n        \\draw[Arete](11)--(10);\n        \\draw[Arete](11)--(12);\n        \\node[Noeud](13)at(13,-1){};\n        \\node[Feuille](14)at(14,-3){};\n        \\node[] (14') [below of = 14] {\\scalebox{3}{$1$}};\n        \\node[Noeud](15)at(15,-2){};\n        \\node[Feuille](16)at(16,-3){};\n        \\draw[Arete](15)--(14);\n        \\draw[Arete](15)--(16);\n        \\draw[Arete](13)--(11);\n        \\draw[Arete](13)--(15);\n        \\draw[Arete](9)--(3);\n        \\draw[Arete](9)--(13);\n    \\end{tikzpicture}}\n    \\caption{The canopy of this binary tree is $0100101$.}\n    \\label{fig:ExempleCanopee}\n\\end{figure}\n\n\\subsection{Baxter permutations and pairs of twin binary trees}\n\n\\subsubsection{Baxter permutations}\nA permutation~$\\sigma$ is a \\emph{Baxter permutation} if for any subword\n$u := u_1 u_2 u_3 u_4$ of~$\\sigma$ such that the letters~$u_2$ and~$u_3$\nare adjacent in~$\\sigma$, $\\operatorname{std}(u) \\notin \\{2413, 3142\\}$. In other\nwords,~$\\sigma$ is a Baxter permutation if it avoids the\n\\emph{generalized permutation patterns} $2-41-3$ and $3-14-2$ (see~\\cite{BS00}\nfor an introduction on generalized permutation patterns). For example,\n${\\bf 4}21{\\bf 73}8{\\bf 5}6$ is not a Baxter permutation; On the other hand\n$436975128$, is a Baxter permutation. Let us denote by~$\\mathfrak{S}^{\\operatorname{B}}_n$ the\nset of Baxter permutations of size~$n$ and by~$\\mathfrak{S}^{\\operatorname{B}}$ the set of all\nBaxter permutations.\n\n\\subsubsection{Pairs of twin binary trees}\nA \\emph{pair of twin binary trees}~$(T_L, T_R)$ is made of two binary trees\n$T_L, T_R \\in \\mathcal{B}\\mathcal{T}_n$ such that the canopies of~$T_L$ and~$T_R$ are\ncomplementary, that is\n\\begin{equation}\n    \\operatorname{cnp}(T_L)_i \\ne \\operatorname{cnp}(T_R)_i \\mbox{ for all } 1 \\leq i \\leq n - 1\n\\end{equation}\n(see Figure~\\ref{fig:ExempleABJ}).\n\\begin{figure}[ht]\n    \\centering\n    \\scalebox{.20}{%\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Feuille](2)at(2,-3){};\n        \\node[] (2') [below of = 2] {\\scalebox{3}{$0$}};\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,-1){};\n        \\node[Feuille](4)at(4,-4){};\n        \\node[] (4') [below of = 4] {\\scalebox{3}{$1$}};\n        \\node[Noeud](5)at(5,-3){};\n        \\node[Feuille](6)at(6,-4){};\n        \\node[] (6') [below of = 6] {\\scalebox{3}{$0$}};\n        \\draw[Arete](5)--(4);\n        \\draw[Arete](5)--(6);\n        \\node[Noeud](7)at(7,-2){};\n        \\node[Feuille](8)at(8,-3){};\n        \\node[] (8') [below of = 8] {\\scalebox{3}{$0$}};\n        \\draw[Arete](7)--(5);\n        \\draw[Arete](7)--(8);\n        \\draw[Arete](3)--(1);\n        \\draw[Arete](3)--(7);\n        \\node[Noeud](9)at(9,0){};\n        \\node[Feuille](10)at(10,-3){};\n        \\node[] (10') [below of = 10] {\\scalebox{3}{$1$}};\n        \\node[Noeud](11)at(11,-2){};\n        \\node[Feuille](12)at(12,-3){};\n        \\node[] (12') [below of = 12] {\\scalebox{3}{$0$}};\n        \\draw[Arete](11)--(10);\n        \\draw[Arete](11)--(12);\n        \\node[Noeud](13)at(13,-1){};\n        \\node[Feuille](14)at(14,-3){};\n        \\node[] (14') [below of = 14] {\\scalebox{3}{$1$}};\n        \\node[Noeud](15)at(15,-2){};\n        \\node[Feuille](16)at(16,-3){};\n        \\draw[Arete](15)--(14);\n        \\draw[Arete](15)--(16);\n        \\draw[Arete](13)--(11);\n        \\draw[Arete](13)--(15);\n        \\draw[Arete](9)--(3);\n        \\draw[Arete](9)--(13);\n    \\end{tikzpicture}}\n    \\enspace\n    \\scalebox{.20}{%\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Feuille](2)at(2,-4){};\n        \\node[] (2') [below of = 2] {\\scalebox{3}{$1$}};\n        \\node[Noeud](3)at(3,-3){};\n        \\node[Feuille](4)at(4,-4){};\n        \\node[] (4') [below of = 4] {\\scalebox{3}{$0$}};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud](5)at(5,-1){};\n        \\node[Feuille](6)at(6,-3){};\n        \\node[] (6') [below of = 6] {\\scalebox{3}{$1$}};\n        \\node[Noeud](7)at(7,-2){};\n        \\node[Feuille](8)at(8,-4){};\n        \\node[] (8') [below of = 8] {\\scalebox{3}{$1$}};\n        \\node[Noeud](9)at(9,-3){};\n        \\node[Feuille](10)at(10,-4){};\n        \\node[] (10') [below of = 10] {\\scalebox{3}{$0$}};\n        \\draw[Arete](9)--(8);\n        \\draw[Arete](9)--(10);\n        \\draw[Arete](7)--(6);\n        \\draw[Arete](7)--(9);\n        \\draw[Arete](5)--(1);\n        \\draw[Arete](5)--(7);\n        \\node[Noeud](11)at(11,0){};\n        \\node[Feuille](12)at(12,-3){};\n        \\node[] (12') [below of = 12] {\\scalebox{3}{$1$}};\n        \\node[Noeud](13)at(13,-2){};\n        \\node[Feuille](14)at(14,-3){};\n        \\node[] (14') [below of = 14] {\\scalebox{3}{$0$}};\n        \\draw[Arete](13)--(12);\n        \\draw[Arete](13)--(14);\n        \\node[Noeud](15)at(15,-1){};\n        \\node[Feuille](16)at(16,-2){};\n        \\draw[Arete](15)--(13);\n        \\draw[Arete](15)--(16);\n        \\draw[Arete](11)--(5);\n        \\draw[Arete](11)--(15);\n    \\end{tikzpicture}}\n    \\caption{A pair of twin binary trees.}\n    \\label{fig:ExempleABJ}\n\\end{figure}\n\\medskip\n\nDenote by~$\\mathcal{T}\\mathcal{B}\\mathcal{T}_n$ the set of pairs of twin binary trees where each\nbinary tree has~$n$ nodes and by~$\\mathcal{T}\\mathcal{B}\\mathcal{T}$ the set of all pairs of twin\nbinary trees.\n\\medskip\n\nAn $A$-labeled pair of twin binary trees~$(T_L, T_R)$ is a \\emph{pair of\ntwin binary search trees} if~$T_L$ (resp.~$T_R$) is an $A$-labeled left\n(resp. right) binary search tree and~$T_L$ and~$T_R$ have the same inorder\nreading. The \\emph{shape}~$\\operatorname{sh}(J)$ of an $A$-labeled pair of twin binary\ntrees $J := (T_L, T_R)$ is the unlabeled pair of twin binary\ntrees~$(\\operatorname{sh}(T_L), \\operatorname{sh}(T_R))$.\n\\medskip\n\nIn~\\cite{DG94}, Dulucq and Guibert have highlighted a bijection between\nBaxter permutations and unlabeled pairs of twin binary trees. In the sequel,\nwe shall make use of a very similar bijection.\n\n\\section{The Baxter monoid} \\label{sec:MonoideBaxter}\n\n\\subsection{Definition and first properties}\nRecall that an equivalence relation~$\\equiv$ defined on~$A^*$ is a \\emph{congruence}\nif for all $u, u', v, v' \\in A^*$, $u \\equiv u'$ and $v \\equiv v'$  imply\n$uv \\equiv u'v'$. Note that the quotient $A^*\/_\\equiv$ of~$A^*$ by a\ncongruence~$\\equiv$ is naturally a monoid. Indeed, by denoting by\n$\\tau : A^* \\to A^*\/_\\equiv$ the canonical projection, the set $A^*\/_\\equiv$\nis endowed with a product~$\\cdot$ defined by\n$\\widehat{u} \\cdot \\widehat{v} := \\tau(uv)$ for all\n$\\widehat{u}, \\widehat{v} \\in A^*\/_\\equiv$ where~$u$ and~$v$ are any words\nsuch that $\\tau(u) = \\widehat{u}$ and $\\tau(v) = \\widehat{v}$.\n\n\\begin{Definition} \\label{def:MonoideBaxter}\n    The \\emph{Baxter monoid} is the quotient of the free monoid~$A^*$ by\n    the congruence~${\\:\\equiv_{\\operatorname{B}}\\:}$ that is the reflexive and transitive closure\n    of the \\emph{Baxter adjacency relations}~${\\:\\leftrightharpoons_{\\operatorname{B}}\\:}$ and~${\\:\\rightleftharpoons_{\\operatorname{B}}\\:}$ defined\n    for $u, v\\in A^*$ and ${\\tt a}, {\\tt b}, {\\tt c}, {\\tt d} \\in A$ by\n    \\begin{align}\n        {\\tt c} \\, u \\, {\\tt a} {\\tt d} \\, v \\, {\\tt b} & {\\:\\leftrightharpoons_{\\operatorname{B}}\\:} {\\tt c} \\, u \\, {\\tt d} {\\tt a} \\, v \\, {\\tt b}\n        \\qquad \\mbox{where \\quad ${\\tt a} \\leq {\\tt b} < {\\tt c} \\leq {\\tt d}$,} \\label{eq:EquivBXAdj1} \\\\\n        {\\tt b} \\, u \\, {\\tt d} {\\tt a} \\, v \\, {\\tt c} & {\\:\\rightleftharpoons_{\\operatorname{B}}\\:} {\\tt b} \\, u \\, {\\tt a} {\\tt d} \\, v \\, {\\tt c}\n        \\qquad \\mbox{where \\quad ${\\tt a} < {\\tt b} \\leq {\\tt c} < {\\tt d}$.} \\label{eq:EquivBXAdj2}\n    \\end{align}\n\\end{Definition}\n\nFor example, the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class of~$2415253$ (see\nFigure~\\ref{fig:ExClasseBaxter}) is\n\\begin{equation}\n    \\{2142553, 2145253, 2145523, 2412553, 2415253, 2415523, 2451253,\n      2451523, 2455123\\}.\n\\end{equation}\n\\begin{figure}[ht]\n    \\centering\n    \\begin{tikzpicture}[scale=.5,font=\\small]\n        \\node (2153674) at (0,0) {$2142553$};\n        \\node (2156374) at (-2,-2) {$2145253$};\n        \\node (2513674) at (2,-2) {$2412553$};\n        \\node (2156734) at (-4,-4) {$2145523$};\n        \\node (2516374) at (0,-4) {$2415253$};\n        \\node (2516734) at (-2,-6) {$2415523$};\n        \\node (2561374) at (2,-6) {$2451253$};\n        \\node (2561734) at (0,-8) {$2451523$};\n        \\node (2567134) at (0,-10) {$2455123$};\n        \\draw [Arete] (2153674) -- (2156374);\n        \\draw [Arete] (2153674) -- (2513674);\n        \\draw [Arete] (2156374) -- (2156734);\n        \\draw [Arete] (2513674) -- (2516374);\n        \\draw [Arete] (2156734) -- (2516734);\n        \\draw [Arete] (2516374) -- (2561374);\n        \\draw [Arete] (2516734) -- (2561734);\n        \\draw [Arete] (2561374) -- (2561734);\n        \\draw [Arete] (2561734) -- (2567134);\n        \\draw [Arete] (2156374) -- (2516374);\n        \\draw [Arete] (2516374) -- (2516734);\n    \\end{tikzpicture}\n    \\qquad\n    \\begin{tikzpicture}[scale=.5,font=\\small]\n        \\node (2153674) at (0,0) {$2153674$};\n        \\node (2156374) at (-2,-2) {$2156374$};\n        \\node (2513674) at (2,-2) {$2513674$};\n        \\node (2156734) at (-4,-4) {$2156734$};\n        \\node (2516374) at (0,-4) {$2516374$};\n        \\node (2516734) at (-2,-6) {$2516734$};\n        \\node (2561374) at (2,-6) {$2561374$};\n        \\node (2561734) at (0,-8) {$2561734$};\n        \\node (2567134) at (0,-10) {$2567134$};\n        \\draw [Arete] (2153674) -- (2156374);\n        \\draw [Arete] (2153674) -- (2513674);\n        \\draw [Arete] (2156374) -- (2156734);\n        \\draw [Arete] (2513674) -- (2516374);\n        \\draw [Arete] (2156734) -- (2516734);\n        \\draw [Arete] (2516374) -- (2561374);\n        \\draw [Arete] (2516734) -- (2561734);\n        \\draw [Arete] (2561374) -- (2561734);\n        \\draw [Arete] (2561734) -- (2567134);\n        \\draw [Arete] (2156374) -- (2516374);\n        \\draw [Arete] (2516374) -- (2516734);\n    \\end{tikzpicture}\n    \\caption{The Baxter equivalence class of the word $u := 2415253$ and of the\n    permutation $2516374 = \\operatorname{std}(u)$. Edges represent Baxter adjacency relations.}\n    \\label{fig:ExClasseBaxter}\n\\end{figure}\n\nNote that if the Baxter congruence is applied on words without repetition,\nthe two Baxter adjacency relations~${\\:\\leftrightharpoons_{\\operatorname{B}}\\:}$ and~${\\:\\rightleftharpoons_{\\operatorname{B}}\\:}$ can be replaced\nby the only adjacency relation~${\\:\\rightleftarrows_{\\operatorname{B}}\\:}$ defined for $u, v \\in A^*$ and\n${\\tt a}, {\\tt b}, {\\tt b}', {\\tt d} \\in A$ by\n\\begin{equation}\n    {\\tt b} \\, u \\, {\\tt a} {\\tt d} \\, v \\, {\\tt b}' {\\:\\rightleftarrows_{\\operatorname{B}}\\:} {\\tt b} \\, u \\, {\\tt d} {\\tt a} \\, v \\, {\\tt b}'\n    \\qquad \\mbox{where \\quad ${\\tt a} < {\\tt b}, {\\tt b}' < {\\tt d}$.}\n\\end{equation}\n\n\\subsubsection{Compatibility with the destandardization process}\nA monoid $A^*\/_\\equiv$ is \\emph{compatible with the destandardization process}\nif for all $u, v \\in A^*$, $u \\equiv v$ if and only if $\\operatorname{std}(u) \\equiv \\operatorname{std}(v)$\nand $\\operatorname{ev}(u) = \\operatorname{ev}(v)$.\n\n\\begin{Proposition} \\label{prop:CompDestd}\n    The Baxter monoid is compatible with the destandardization process.\n\\end{Proposition}\n\\begin{proof}\n    It is enough to check the property on adjacency relations. Let\n    $u, v \\in A^*$. Assume $u {\\:\\leftrightharpoons_{\\operatorname{B}}\\:} v$. We have\n    \\begin{equation}\n        u = x \\, {\\tt c} \\, y \\, {\\tt a} {\\tt d} \\, z \\, {\\tt b} \\, t\n        \\quad \\mbox{and} \\quad\n        v = x \\, {\\tt c} \\, y \\, {\\tt d} {\\tt a} \\, z \\, {\\tt b} \\, t\n    \\end{equation}\n    for some letters ${\\tt a} \\leq {\\tt b} < {\\tt c} \\leq {\\tt d}$ and words~$x$, $y$, $z$,\n    and~$t$. Since~${\\:\\leftrightharpoons_{\\operatorname{B}}\\:}$ acts by permuting letters, we have\n    $\\operatorname{ev}(u) = \\operatorname{ev}(v)$. Moreover, the letters~${\\tt a}'$, ${\\tt b}'$, ${\\tt c}'$\n    and~${\\tt d}'$ of~$\\operatorname{std}(u)$ respectively at the same positions as the\n    letters~${\\tt a}$, ${\\tt b}$, ${\\tt c}$ and~${\\tt d}$ of~$u$ satisfy\n    ${\\tt a}' < {\\tt b}' < {\\tt c}' < {\\tt d}'$ due to their relative positions into~$\\operatorname{std}(u)$\n    and the order relations between~${\\tt a}$, ${\\tt b}$, ${\\tt c}$ and~${\\tt d}$. The same\n    relations hold for the letters of~$\\operatorname{std}(v)$, showing that\n    $\\operatorname{std}(u) {\\:\\leftrightharpoons_{\\operatorname{B}}\\:} \\operatorname{std}(v)$. The proof is analogous for the case $u {\\:\\rightleftharpoons_{\\operatorname{B}}\\:} v$.\n\n    Conversely, assume that~$v$ is a permutation of~$u$ and $\\operatorname{std}(u) {\\:\\leftrightharpoons_{\\operatorname{B}}\\:} \\operatorname{std}(v)$.\n    We have\n    \\begin{equation}\n        \\operatorname{std}(u) = x \\, {\\tt c} \\, y \\, {\\tt a} {\\tt d} \\, z \\, {\\tt b} \\, t\n        \\quad \\mbox{and} \\quad\n        \\operatorname{std}(v) = x \\, {\\tt c} \\, y \\, {\\tt d} {\\tt a} \\, z \\, {\\tt b} \\, t\n    \\end{equation}\n    for some letters ${\\tt a} < {\\tt b} < {\\tt c} < {\\tt d}$ and words~$x$, $y$, $z$, and~$t$.\n    The word~$u$ is a non-standardized version of~$\\operatorname{std}(u)$ so that the\n    letters~${\\tt a}'$, ${\\tt b}'$, ${\\tt c}'$ and~${\\tt d}'$ of~$u$ respectively at the\n    same positions as the letters~${\\tt a}$, ${\\tt b}$, ${\\tt c}$ and~${\\tt d}$ of~$\\operatorname{std}(u)$\n    satisfy ${\\tt a}' \\leq {\\tt b}' < {\\tt c}' \\leq {\\tt d}'$ due to their relative positions\n    into~$u$ and the order relations between~${\\tt a}$, ${\\tt b}$, ${\\tt c}$ and~${\\tt d}$.\n    The same relations hold for the letters of~$v$, showing that $u {\\:\\leftrightharpoons_{\\operatorname{B}}\\:} v$.\n    The proof is analogous for the case $\\operatorname{std}(u) {\\:\\rightleftharpoons_{\\operatorname{B}}\\:} \\operatorname{std}(v)$.\n\\end{proof}\n\n\\subsubsection{Compatibility with the restriction of alphabet intervals}\nA monoid $A^*\/_\\equiv$ is \\emph{compatible with the restriction of alphabet\nintervals} if for any interval~$I$ of~$A$ and for all $u, v \\in A^*$,\n$u \\equiv v$ implies $u_{|I} \\equiv v_{|I}$.\n\n\\begin{Proposition} \\label{prop:CompRestrSegmAlph}\n    The Baxter monoid is compatible with the restriction of alphabet\n    intervals.\n\\end{Proposition}\n\\begin{proof}\n    It is enough to check the property on adjacency relations. Moreover,\n    by Proposition~\\ref{prop:CompDestd}, it is enough to check the property\n    for permutations. Let $\\sigma, \\nu \\in \\mathfrak{S}_n$ such that $\\sigma {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu$.\n    We have $\\sigma = x \\, {\\tt b} \\, y \\, {\\tt a} {\\tt d} \\, z \\, {\\tt b}' \\, t$ and\n    $\\nu = x \\, {\\tt b} \\, y \\, {\\tt d} {\\tt a} \\, z \\, {\\tt b}' \\, t$ for some letters\n    ${\\tt a} < {\\tt b}, {\\tt b}' < {\\tt d}$ and words $x$, $y$, $z$, and~$t$. Let~$I$ be\n    an interval of $\\{1, \\dots, n\\}$ and $R := I \\cap \\{{\\tt a}, {\\tt b}, {\\tt b}', {\\tt d}\\}$.\n    If $R = \\{{\\tt a}, {\\tt b}, {\\tt b}', {\\tt d}\\}$,\n    \\begin{equation}\n        \\sigma_{|I} = x_{|I} \\, {\\tt b} \\, y_{|I} \\, {\\tt a} {\\tt d} \\, z_{|I} \\, {\\tt b}' \\, t_{|I}\n        \\quad \\mbox{and} \\quad\n        \\nu_{|I} = x_{|I} \\, {\\tt b} \\, y_{|I} \\, {\\tt d} {\\tt a} \\, z_{|I} \\, {\\tt b}' \\, t_{|I}\n    \\end{equation}\n    so that $\\sigma_{|I} {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu_{|I}$. Otherwise, we have\n    $\\sigma_{|I} = \\nu_{|I}$ and thus $\\sigma_{|I} {\\:\\equiv_{\\operatorname{B}}\\:} \\nu_{|I}$.\n\\end{proof}\n\n\\subsubsection{Compatibility with the Sch\u00fctzenberger involution}\nA monoid $A^*\/_\\equiv$ is \\emph{compatible with the Sch\u00fctzenberger involution}\nif for all $u, v \\in A^*$, $u \\equiv v$ implies $u^\\# \\equiv v^\\#$.\n\n\\begin{Proposition} \\label{prop:CompSchutz}\n    The Baxter monoid is compatible with the Sch\u00fctzenberger involution.\n\\end{Proposition}\n\\begin{proof}\n    It is enough to check the property on adjacency relations. Moreover, by\n    Proposition~\\ref{prop:CompDestd}, it is enough to check the property\n    for permutations. Let $\\sigma, \\nu \\in \\mathfrak{S}_n$ and assume that\n    $\\sigma {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu$. We have $\\sigma = x \\, {\\tt b} \\, y \\, {\\tt a} {\\tt d} \\, z \\, {\\tt b}' \\, t$\n    and $\\nu = x \\, {\\tt b} \\, y \\, {\\tt d} {\\tt a} \\, z \\, {\\tt b}' \\, t$ for some letters\n    ${\\tt a} < {\\tt b}, {\\tt b}' < {\\tt d}$ and words~$x$, $y$, $z$, and~$t$. We have\n    \\begin{equation}\n        \\sigma^\\# = t^\\# \\, {\\tt b}'^\\# \\, z^\\# \\, {\\tt d}^\\# {\\tt a}^\\# \\, y^\\# \\, {\\tt b}^\\# \\, x^\\#\n        \\quad \\mbox{and} \\quad\n        \\nu^\\# = t^\\# \\, {\\tt b}'^\\# \\, z^\\# \\, {\\tt a}^\\# {\\tt d}^\\# \\, y^\\# \\, {\\tt b}^\\# \\, x^\\#.\n    \\end{equation}\n    Since ${\\tt d}^\\# < {\\tt b}'^\\#, {\\tt b}^\\# < {\\tt a}^\\#$, we have $\\sigma^\\# {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu^\\#$.\n\\end{proof}\n\n\\subsection{Connection with the sylvester monoid}\nThe \\emph{sylvester monoid}~\\cite{HNT02, HNT05} is the quotient of the\nfree monoid~$A^*$ by the congruence~${\\:\\equiv_{\\operatorname{S}}\\:}$ that is the reflexive and\ntransitive closure of the \\emph{sylvester adjacency relation}~${\\:\\leftrightharpoons_{\\operatorname{S}}\\:}$\ndefined for $u \\in A^*$ and ${\\tt a}, {\\tt b}, {\\tt c} \\in A$ by\n\\begin{equation}\n    {\\tt a} {\\tt c} \\, u \\, {\\tt b} {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} {\\tt c} {\\tt a} \\, u \\, {\\tt b}\n    \\qquad \\mbox{where \\quad ${\\tt a} \\leq {\\tt b} < {\\tt c}$.}\n\\end{equation}\nIn the same way, let us define the \\emph{$\\#$-sylvester monoid}, the quotient\nof~$A^*$ by the congruence~${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$ that is the reflexive and transitive\nclosure of the \\emph{$\\#$-sylvester adjacency relation}~${\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:}$ defined\nfor $u\\in A^*$ and ${\\tt a}, {\\tt b}, {\\tt c} \\in A$ by\n\\begin{equation} \\label{eq:DefSchutzS}\n    {\\tt b} \\, u \\, {\\tt a} {\\tt c} {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} {\\tt b} \\, u \\, {\\tt c} {\\tt a}\n    \\qquad \\mbox{where \\quad ${\\tt a} < {\\tt b} \\leq {\\tt c}$.}\n\\end{equation}\nNote that this adjacency relation is defined by taking the images by the\nSch\u00fctzenberger involution of the sylvester adjacency relation. Indeed, for\nall $u, v \\in A^*$, $u {\\:\\equiv_{\\operatorname{S}^\\#}\\:} v$ if and only if~$u^\\# {\\:\\equiv_{\\operatorname{S}}\\:} v^\\#$.\n\\medskip\n\nIn~\\cite{HNT05}, Hivert, Novelli and Thibon have shown that two words\nare sylvester equivalent if and only if each gives the same right binary\nsearch tree by inserting their letters from right to left using the binary\nsearch tree insertion algorithm~\\cite{AU94}. In our setting, we call this\nprocess the \\emph{leaf insertion} and it comes in two versions, depending\non if the considered binary tree is a right or a left binary search tree:\n\\medskip\n\n{\\flushleft\n    {\\bf Algorithm:} {\\sc LeafInsertion}. \\\\\n    {\\bf Input:} An $A$-labeled right (resp. left) binary search tree~$T$,\n    a letter~${\\tt a} \\in A$. \\\\\n    {\\bf Output:} $T$ after the leaf insertion of~${\\tt a}$. \\\\\n    \\begin{enumerate}\n        \\item If $T = \\perp$, return the one-node binary search tree\n        labeled by~${\\tt a}$.\n        \\item Let~${\\tt b}$ be the label of the root of~$T$.\n        \\item If ${\\tt a} \\leq {\\tt b}$ (resp. ${\\tt a} < {\\tt b}$): \\label{item:InstrDiff}\n        \\begin{enumerate}\n            \\item Then, recursively leaf insert~${\\tt a}$ into the left subtree\n            of~$T$.\n            \\item Otherwise, recursively leaf insert ${\\tt a}$ into the right\n            subtree of~$T$.\n        \\end{enumerate}\n    \\end{enumerate}\n    {\\bf End.}\n}\n\\medskip\n\nFor further reference, let us recall the following theorem due to Hivert,\nNovelli and Thibon~\\cite{HNT05}, restated in our setting and supplemented\nwith a respective part:\n\\begin{Theoreme} \\label{thm:PSymbPBT}\n    Two words are ${\\:\\equiv_{\\operatorname{S}}\\:}$-equivalent (resp. ${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$-equivalent)\n    if and only if they give the same right (resp. left) binary search tree\n    by inserting their letters from right to left (resp. left to right).\n\\end{Theoreme}\n\nIn other words, any $A$-labeled right (resp. left) binary search tree\nencodes a sylvester (resp. $\\#$-sylvester) equivalence class of words\nof~$A^*$, and conversely.\n\\smallskip\n\nLet us explain the respective part of Theorem~\\ref{thm:PSymbPBT}. It follows\nfrom~(\\ref{eq:DefSchutzS}) that encoding the~${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$-equivalence\nclass of a word~$u$ is equivalent to encoding the~${\\:\\equiv_{\\operatorname{S}}\\:}$-equivalence\nclass of~$u^\\#$. For this, simply insert~$u$ from left to right by considering\nthat the reversed order relation holds between its letters. In this way,\nwe obtain a binary tree such that for any node~$x$ labeled by a letter~${\\tt b}$,\nall labels~${\\tt a}$ of the nodes of the left subtree of~$x$, and all labels~${\\tt c}$\nof the nodes of the right subtree of~$x$, the inequality ${\\tt a} \\geq {\\tt b} > {\\tt c}$\nholds. This binary tree is obviously not a left binary search tree. Nevertheless,\na left binary search tree can be obtained from it after swapping, for each\nnode, its left and right subtree recursively. One can prove by induction\non~$|u|$ that this left binary search tree is the one that {\\sc LeafInsertion}\nconstructs by inserting the letters of~$u$ from left to right and hence,\nthis remark explains the difference of treatment between right and left\nbinary search trees for the instruction~(\\ref{item:InstrDiff}) of\n{\\sc LeafInsertion}.\n\n\\begin{Lemme} \\label{lem:DiagHasseEqS}\n    Let $u := x \\, {\\tt a} {\\tt c} \\, y$ and $v := x \\, {\\tt c} {\\tt a} \\, y$ be two words\n    such that $x$ and $y$ are two words, ${\\tt a} < {\\tt c}$ are two letters,\n    and~$u {\\:\\equiv_{\\operatorname{S}}\\:} v$. Then,~$u {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} v$.\n\\end{Lemme}\n\\begin{proof}\n    Follows from Theorem~\\ref{thm:PSymbPBT}: Since~$u$ and~$v$ give the\n    same right binary search tree~$T$ by inserting these from right to left,\n    the node labeled by~${\\tt a}$ and the node labeled by~${\\tt c}$ in~$T$ cannot\n    be ancestor one of the other. That implies that there exists a node\n    labeled by a letter~${\\tt b}$, common ancestor of both nodes labeled by~${\\tt a}$\n    and~${\\tt c}$ such that ${\\tt a} \\leq {\\tt b} < {\\tt c}$. Thus,~$u {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} v$.\n\\end{proof}\n\nLemma~\\ref{lem:DiagHasseEqS} also proves that the ${\\:\\leftrightharpoons_{\\operatorname{S}}\\:}$-adjacency\nrelations of any equivalence class~$C$ of $\\mathfrak{S}_n \/_{\\:\\equiv_{\\operatorname{S}}\\:}$ are\nexactly the covering relations of the permutohedron restricted to the\nelements of~$C$. Note that it is also the case for the ${\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:}$-adjacency\nrelations.\n\\medskip\n\nThe Baxter monoid, the sylvester monoid and the $\\#$-sylvester monoid are\nrelated in the following way.\n\\begin{Proposition} \\label{prop:LienSylv}\n    Let $u, v \\in A^*$. Then, $u {\\:\\equiv_{\\operatorname{B}}\\:} v$ if and only if $u {\\:\\equiv_{\\operatorname{S}}\\:} v$\n    and $u {\\:\\equiv_{\\operatorname{S}^\\#}\\:} v$.\n\\end{Proposition}\n\\begin{proof}\n    $(\\Rightarrow)$: Once more, it is enough to check the property on adjacency\n    relations. Moreover, by Proposition~\\ref{prop:CompDestd}, it is enough\n    to check the property for permutations. Let $\\sigma, \\nu \\in \\mathfrak{S}_n$\n    and assume that~$\\sigma {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu$. We have\n    $\\sigma = x \\, {\\tt b} \\, y \\, {\\tt a} {\\tt d} \\, z \\, {\\tt b}' \\, t$\n    and $\\nu = x \\, {\\tt b} \\, y \\, {\\tt d} {\\tt a} \\, z \\, {\\tt b}' \\, t$ for some letters\n    ${\\tt a} < {\\tt b}, {\\tt b}' < {\\tt d}$ and words~$x$, $y$, $z$, and~$t$. The presence\n    of the letters~${\\tt a}$, ${\\tt d}$ and~${\\tt b}'$ with ${\\tt a} < {\\tt b}' < {\\tt d}$ ensures\n    that~$\\sigma {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} \\nu$. Besides, the presence of the letters~${\\tt b}$,\n    ${\\tt a}$ and~${\\tt d}$ with ${\\tt a} < {\\tt b} < {\\tt d}$ ensures that~$\\sigma {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} \\nu$.\n\n    $(\\Leftarrow)$: Since the sylvester and the $\\#$-sylvester monoids are\n    compatible with the destandardization process~\\cite{HNT05}, it is enough\n    to check the property for permutations. Let $\\sigma, \\nu \\in \\mathfrak{S}_n$\n    such that~$\\sigma {\\:\\equiv_{\\operatorname{S}}\\:} \\nu$ and~$\\sigma {\\:\\equiv_{\\operatorname{S}^\\#}\\:} \\nu$. Set\n    $\\tau := \\inf_{{\\:\\leq_{\\operatorname{P}}\\:}} \\{\\sigma, \\nu\\}$. Since the permutohedron\n    is a lattice,~$\\tau$ is well-defined, and since the equivalence classes\n    of permutations under the~${\\:\\equiv_{\\operatorname{S}}\\:}$ and~${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$ congruences\n    are intervals of the permutohedron~\\cite{HNT05}, we have\n    $\\sigma {\\:\\equiv_{\\operatorname{S}}\\:} \\tau {\\:\\equiv_{\\operatorname{S}}\\:} \\nu$ and\n    $\\sigma {\\:\\equiv_{\\operatorname{S}^\\#}\\:} \\tau {\\:\\equiv_{\\operatorname{S}^\\#}\\:} \\nu$. Moreover, by\n    Lemma~\\ref{lem:DiagHasseEqS}, and again since that the equivalence\n    classes of permutations under the~${\\:\\equiv_{\\operatorname{S}}\\:}$ and the~${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$\n    congruences are intervals of the permutohedron, for each saturated\n    chains $\\tau {\\:\\leq_{\\operatorname{P}}\\:} \\sigma' {\\:\\leq_{\\operatorname{P}}\\:} \\cdots {\\:\\leq_{\\operatorname{P}}\\:} \\sigma$\n    and $\\tau {\\:\\leq_{\\operatorname{P}}\\:} \\nu' {\\:\\leq_{\\operatorname{P}}\\:} \\cdots {\\:\\leq_{\\operatorname{P}}\\:} \\nu$, there are\n    sequences of adjacency relations $\\tau {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} \\sigma' {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} \\cdots {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} \\sigma$,\n    $\\tau {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} \\sigma' {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} \\cdots {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} \\sigma$,\n    $\\tau {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} \\nu' {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} \\cdots {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} \\nu$ and\n    $\\tau {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} \\nu' {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} \\cdots {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} \\nu$. Hence,\n    $\\tau {\\:\\equiv_{\\operatorname{B}}\\:} \\sigma$ and $\\tau {\\:\\equiv_{\\operatorname{B}}\\:} \\nu$, implying~$\\sigma {\\:\\equiv_{\\operatorname{B}}\\:} \\nu$.\n\\end{proof}\n\nProposition~\\ref{prop:LienSylv} shows that the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\nclasses are the intersection of ${\\:\\equiv_{\\operatorname{S}}\\:}$-equivalence classes and\n${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$-equivalence classes.\n\\medskip\n\nBy the characterization of the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes provided by\nProposition~\\ref{prop:LienSylv}, restricting the Baxter congruence on\npermutations, we have the following property:\n\\begin{Proposition} \\label{prop:EquivBXInter}\n    For any $n \\geq 0$, each equivalence class of $\\mathfrak{S}_n \/_{\\:\\equiv_{\\operatorname{B}}\\:}$\n    is an interval of the permutohedron.\n\\end{Proposition}\n\\begin{proof}\n    By Proposition~\\ref{prop:LienSylv}, the~${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes\n    are the intersection of the~${\\:\\equiv_{\\operatorname{S}}\\:}$ and the~${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$-equivalence\n    classes. Moreover, the permutations under the~${\\:\\equiv_{\\operatorname{S}}\\:}$ and the~${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$\n    equivalence relations are intervals of the permutohedron~\\cite{HNT05}.\n    The proposition comes from the fact that the intersection of two lattice\n    intervals is also an interval and that the permutohedron is a lattice.\n\\end{proof}\n\n\\begin{Lemme} \\label{lem:DiagHasseEqBX}\n    Let $u := x \\, {\\tt a} {\\tt d} \\, y$ and $v := x \\, {\\tt d} {\\tt a} \\, y$\n    such that $x$ and $y$ are two words, ${\\tt a} < {\\tt d}$ are two letters,\n    and~$u {\\:\\equiv_{\\operatorname{B}}\\:} v$. Then,~$u {\\:\\leftrightharpoons_{\\operatorname{B}}\\:} v$ or~$u {\\:\\rightleftharpoons_{\\operatorname{B}}\\:} v$.\n\\end{Lemme}\n\\begin{proof}\n    By Proposition~\\ref{prop:LienSylv}, since~$u {\\:\\equiv_{\\operatorname{B}}\\:} v$, we\n    have~$u {\\:\\equiv_{\\operatorname{S}}\\:} v$ and thus by Lemma~\\ref{lem:DiagHasseEqS} we\n    have~$u {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} v$, implying the existence of a letter~${\\tt b}'$ in the\n    factor~$y$ satisfying ${\\tt a} \\leq {\\tt b}' < {\\tt d}$. In the same way, we also\n    have~$u {\\:\\equiv_{\\operatorname{S}^\\#}\\:} v$ and thus~$u {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} v$, hence the existence\n    of a letter~${\\tt b}$ in the factor~$x$ satisfying ${\\tt a} < {\\tt b} \\leq {\\tt d}$.\n    That proves that~$u$ and~$v$ are~${\\:\\leftrightharpoons_{\\operatorname{B}}\\:}$ or~${\\:\\rightleftharpoons_{\\operatorname{B}}\\:}$-adjacent.\n\\end{proof}\n\nLemma~\\ref{lem:DiagHasseEqBX} is the analog, in the case of the Baxter\ncongruence, of Lemma~\\ref{lem:DiagHasseEqS} and also proves that the~${\\:\\leftrightharpoons_{\\operatorname{B}}\\:}$\nand ${\\:\\rightleftharpoons_{\\operatorname{B}}\\:}$-adjacency relations of any equivalence class~$C$ of\n$\\mathfrak{S}_n \/_{\\:\\equiv_{\\operatorname{B}}\\:}$ are exactly the covering relations of the\npermutohedron restricted to the elements of~$C$.\n\n\\subsection{\\texorpdfstring{Connection with the $3$-recoil monoid}\n                           {Connection with the 3-recoil monoid}}\nIf~${\\tt a}$ and~${\\tt c}$ are two letters of~$A$, denote by~${\\tt c} - {\\tt a}$ the\ncardinality of the set $\\{{\\tt b} \\in A : {\\tt a} < {\\tt b} \\leq {\\tt c}\\}$.\nIn~\\cite{NRT11}, Novelli, Reutenauer and Thibon defined for any~$k \\geq 0$\nthe congruence~$\\EquivR{k}$. This congruence is the reflexive and transitive\nclosure of the \\emph{$k$-recoil adjacency relation}, defined for\n${\\tt a}, {\\tt b} \\in A$ by\n\\begin{equation}\n    {\\tt a} {\\tt b} \\AdjR{k} {\\tt b} {\\tt a} \\qquad \\mbox{where \\quad ${\\tt b} - {\\tt a} \\geq k$.}\n\\end{equation}\nThe \\emph{$k$-recoil monoid} is the quotient of the free monoid~$A^*$ by\nthe congruence~$\\EquivR{k}$. Note that the congruence~$\\EquivR{2}$ restricted\nto permutations is nothing but the \\emph{hypoplactic congruence}~\\cite{N98}.\n\\medskip\n\nThe Baxter monoid and the $3$-recoil monoid are related in the following way.\n\\begin{Proposition} \\label{prop:Lien3Recul}\n    Each~$\\EquivR{3}$-equivalence class of permutations can be expressed\n    as a union of some~${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes.\n\\end{Proposition}\n\\begin{proof}\n    This amounts to prove that for all permutations~$\\sigma$ and~$\\nu$,\n    if~$\\sigma {\\:\\equiv_{\\operatorname{B}}\\:} \\nu$ then~$\\sigma \\EquivR{3} \\nu$. It is enough\n    to check this property on adjacency relations. Hence, assume\n    that~$\\sigma {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu$. We have\n    $\\sigma = x \\, {\\tt b} \\, y \\, {\\tt a} {\\tt d} \\, z {\\tt b}' \\, t$ and\n    $\\nu = x \\, {\\tt b} \\, y \\, {\\tt d} {\\tt a} \\, z \\, {\\tt b}' \\, t$ for some letters\n    ${\\tt a} < {\\tt b}, {\\tt b}' < {\\tt d}$ and words~$x$, $y$, $z$, and~$t$.\n    Since~$\\sigma$ and~$\\nu$ are permutations, ${\\tt b} \\ne {\\tt b}'$ and thus,\n    we have ${\\tt a} < {\\tt b} < {\\tt b}' < {\\tt d}$ or ${\\tt a} < {\\tt b}' < {\\tt b} < {\\tt d}$,\n    implying that~${\\tt d} - {\\tt a} \\geq 3$. Hence,~$\\sigma \\EquivR{3} \\nu$.\n\\end{proof}\n\nNote that Proposition~\\ref{prop:Lien3Recul} is false for the congruence~$\\EquivR{4}$\nsince there are twenty-two equivalence classes of permutations of size~$4$\nunder the congruence~${\\:\\equiv_{\\operatorname{B}}\\:}$ but twenty-four under~$\\EquivR{4}$. Conversely,\nnote that~$\\EquivR{4}$ is not a refinement of~${\\:\\equiv_{\\operatorname{B}}\\:}$ since for any~$n \\geq 5$,\nthe permutation $1 . n . n\\!-\\!1 \\dots 2$ is the only member of\nits~${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class but not of its~$\\EquivR{4}$-equivalence class.\n\\medskip\n\nMoreover, it is clear, by definition of~$\\EquivR{k}$, that the~$\\EquivR{k}$-equivalence\nclasses of permutations are union of~$\\EquivR{k + 1}$-equivalence classes.\nHence, by Proposition~\\ref{prop:Lien3Recul}, the hypoplactic equivalence\nclasses of permutations are union of some~${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes.\n\n\\section{A Robinson-Schensted-like algorithm} \\label{sec:RobinsonSchensted}\nThe goal of this section is to define an analog to the Robinson-Schensted\nalgorithm for the Baxter monoid---see~\\cite{LS81,Lot02} for the usual Robinson-Schensted insertion algorithm that associate to any word $u$ its\n${\\sf P}$-symbol, that is a Young tableau.\n\\medskip\n\nThe interest of the Baxter monoid in our context is that the equivalence\nclasses of the permutations of size~$n$ under the Baxter congruence are\nequinumerous with unlabeled pairs of twin binary trees with~$n$ nodes,\nand thus, by the results of Dulucq and Guibert~\\cite{DG94}, also equinumerous\nwith Baxter permutations of size~$n$. We shall provide a proof of this\nproperty in this section, using our analog of the Robinson-Schensted algorithm.\n\n\\subsection{Principle of the algorithm} \\label{subsec:PSymbBaxter}\nWe describe here an algorithm testing if two words are equivalent\naccording to the Baxter congruence. Given a word~$u \\in A^*$, it computes\nits \\emph{Baxter ${\\sf P}$-symbol}, that is an $A$-labeled pair $(T_L, T_R)$\nconsisting in a left and a right binary search tree such that the nondecreasing\nrearrangement of~$u$ is the inorder reading of both~$T_L$ and~$T_R$. It also\ncomputes its \\emph{Baxter ${\\sf Q}$-symbol}, that is a pair of twin binary\ntrees $(S_L, S_R)$ where~$S_L$ (resp.~$S_R$) is an increasing (resp. decreasing)\nbinary tree, such that the inorder reading of~$S_L$ and~$S_R$ are the same.\nMoreover,~$T_L$ and~$S_L$ have same shape, and so have~$T_R$ and~$S_R$.\n\n\\subsubsection{\\texorpdfstring{The Baxter ${\\sf P}$-symbol}{The Baxter P-symbol}}\n\n\\begin{Definition} \\label{def:BaxterPSymb}\n    The \\emph{Baxter ${\\sf P}$-symbol} (or simply \\emph{${\\sf P}$-symbol}\n    if the context is clear) of a word $u \\in A^*$ is the pair\n    ${\\sf P}(u) = (T_L, T_R)$ where~$T_L$ (resp.~$T_R$) is the left (resp.\n    right) binary search tree obtained by leaf inserting the letters of~$u$\n    from left to right (resp. right to left).\n\\end{Definition}\nFigure~\\ref{fig:ExemplePQSymboleSansEtapes} shows the ${\\sf P}$-symbol of\n$u := 2415253$. Before showing that the ${\\sf P}$-symbol of\nDefinition~\\ref{def:BaxterPSymb} can be used to decide if two words are\nequivalent under the Baxter congruence, let us give an intuitive explanation\nof its validity.\n\\medskip\n\nRecall that, according to Proposition~\\ref{prop:LienSylv}, to represent\nthe Baxter equivalence class of a word~$u$, one has to represent both the\nequivalence class of~$u$ under the~${\\:\\equiv_{\\operatorname{S}}\\:}$ congruence and the equivalence\nclass of~$u$ under the~${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$ congruence. This is exactly what\nthe Baxter ${\\sf P}$-symbol does since, for a word~$u$, it computes a\npair~$(T_L, T_R)$ where, by Theorem~\\ref{thm:PSymbPBT},~$T_L$ represents\nthe~${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$-equivalence class of~$u$ and~$T_R$ represents\nthe~${\\:\\equiv_{\\operatorname{S}}\\:}$-equivalence class of~$u$.\n\n\\subsubsection{\\texorpdfstring{The Baxter ${\\sf Q}$-symbol}{The Baxter Q-symbol}}\nLet us first recall two algorithms. Let~$u$ be a word. Define~$\\operatorname{incr}(u)$,\nthe \\emph{increasing binary tree of~$u$} recursively by\n\\begin{equation}\n    \\operatorname{incr}(u) :=\n    \\begin{cases}\n        \\perp & \\mbox{if $u = \\epsilon$,} \\\\[.5em]\n        \\operatorname{incr}(v) \\wedge_{\\tt a} \\operatorname{incr}(w) &\n        \\mbox{where $u = v {\\tt a} w$, ${\\tt a} = \\min(u)$, and ${\\tt a} < \\min(v)$.}\n    \\end{cases}\n\\end{equation}\nIn the same way, define the \\emph{decreasing binary tree of~$u$}~$\\operatorname{decr}(u)$, by\n\\begin{equation}\n    \\operatorname{decr}(u) :=\n    \\begin{cases}\n        \\perp & \\mbox{if $u = \\epsilon$,} \\\\[.5em]\n        \\operatorname{decr}(v) \\wedge_{\\tt b} \\operatorname{decr}(w) &\n        \\mbox{where $u = v {\\tt b} w$, ${\\tt b} = \\max(u)$, and ${\\tt b} > \\max(w)$.}\n    \\end{cases}\n\\end{equation}\n\n\\begin{Definition} \\label{def:BaxterQSymbole}\n    The \\emph{Baxter ${\\sf Q}$-symbol} (or simply \\emph{${\\sf Q}$-symbol}\n    if the context is clear) of a word $u \\in A^*$ is the pair\n    ${\\sf Q}(u) = (S_L, S_T)$ where\n    \\begin{equation}\n        S_L := \\operatorname{incr}\\left(\\operatorname{std}(u)^{-1}\\right)\n        \\qquad \\mbox{and} \\qquad\n        S_R := \\operatorname{decr}\\left(\\operatorname{std}(u)^{-1}\\right).\n    \\end{equation}\n\\end{Definition}\n\nFigure~\\ref{fig:ExemplePQSymboleSansEtapes} shows the ${\\sf Q}$-symbol of\n$u := 2415253$, whose standardized word is $2516374$, so that\n$\\operatorname{std}(u)^{-1} = 3157246$.\n\\begin{figure}[ht]\n    \\centering\n    \\begin{equation*}\n    \\begin{split}{\\sf P}(u) = \\: \\end{split}\n    \\begin{split}\n    \\scalebox{.34}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$1$};\n        \\node[Noeud,EtiqClair](1)at(1.0,0){$2$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2.0,-2){$2$};\n        \\node[Noeud,EtiqClair](3)at(3.0,-3){$3$};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud,EtiqClair](4)at(4.0,-1){$4$};\n        \\draw[Arete](4)--(2);\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$5$};\n        \\node[Noeud,EtiqClair](6)at(6.0,-3){$5$};\n        \\draw[Arete](5)--(6);\n        \\draw[Arete](4)--(5);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.34}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-2){$1$};\n        \\node[Noeud,EtiqClair](1)at(1.0,-3){$2$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqClair](2)at(2.0,-1){$2$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqClair](3)at(3.0,0){$3$};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud,EtiqClair](4)at(4.0,-3){$4$};\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$5$};\n        \\draw[Arete](5)--(4);\n        \\node[Noeud,EtiqClair](6)at(6.0,-1){$5$};\n        \\draw[Arete](6)--(5);\n        \\draw[Arete](3)--(6);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\qquad\n    \\begin{split}{\\sf Q}(u) = \\: \\end{split}\n    \\begin{split}\n    \\scalebox{.34}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$3$};\n        \\node[Noeud,EtiqClair](1)at(1.0,0){$1$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2.0,-2){$5$};\n        \\node[Noeud,EtiqClair](3)at(3.0,-3){$7$};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud,EtiqClair](4)at(4.0,-1){$2$};\n        \\draw[Arete](4)--(2);\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$4$};\n        \\node[Noeud,EtiqClair](6)at(6.0,-3){$6$};\n        \\draw[Arete](5)--(6);\n        \\draw[Arete](4)--(5);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.34}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-2){$3$};\n        \\node[Noeud,EtiqClair](1)at(1.0,-3){$1$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqClair](2)at(2.0,-1){$5$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqClair](3)at(3.0,0){$7$};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud,EtiqClair](4)at(4.0,-3){$2$};\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$4$};\n        \\draw[Arete](5)--(4);\n        \\node[Noeud,EtiqClair](6)at(6.0,-1){$6$};\n        \\draw[Arete](6)--(5);\n        \\draw[Arete](3)--(6);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\end{equation*}\n    \\caption{The ${\\sf P}$-symbol and the ${\\sf Q}$-symbol of $u := 2415253$.}\n    \\label{fig:ExemplePQSymboleSansEtapes}\n\\end{figure}\n\\medskip\n\nIt is plain that given a word~$u$, the ${\\sf Q}$-symbol of~$u$ allows, in\naddition with its ${\\sf P}$-symbol, to retrieve the original word. Indeed,\nif ${\\sf P}(u) = (T_L, T_R)$ and ${\\sf Q}(u) = (S_L, S_R)$, the pair~$(T_R, S_R)$\nis the output of the Robinson-Schensted-like algorithm in the context of\nthe sylvester monoid, which is a bijection between words and pairs of such\nbinary trees~\\cite{HNT05}. Given~$(T_R, S_R)$, it amounts to reading the\nlabels of~$T_R$ in the order of the corresponding labels in~$S_R$. The\nsame holds of the pair~$(T_L, S_L)$.\n\n\\subsection{Correctness of the insertion algorithm}\n\\begin{Lemme} \\label{lem:OrientationFeuille}\n    Let~$T$ be a non-empty binary tree and~$y$ be the $i$-th leaf of~$T$.\n    If~$y$ is left-oriented, it is attached to the $i$-th node of~$T$.\n    If~$y$ is right-oriented, it is attached to the $i\\!-\\!1$-st node of~$T$.\n\\end{Lemme}\n\\begin{proof}\n    We proceed by structural induction on the set of non-empty binary trees.\n    If~$T$ is the one-node  binary tree, the lemma is clearly satisfied.\n    Otherwise, we have $T = A \\wedge B$. Let~$y$ be the $i$-th leaf of~$T$\n    and~$x$ be the node where~$y$ is attached. If~$y$ is also in~$A$ and\n    $A = \\perp$, $y$ is left-oriented and is attached to the root\n    of~$T$ (that is the first node of~$T$) and the lemma is satisfied. If~$y$\n    is in~$A$ and $A \\ne \\perp$, $y$ is also the $i$-th leaf of~$A$\n    and~$x$ is a node of~$A$, so that the lemma follows by induction hypothesis\n    on~$A$. Otherwise,~$y$ is in~$B$. If $B = \\perp$, $y$ is right-oriented\n    and is attached to the root of~$T$ (that is the last node of~$T$) and\n    the lemma is satisfied. Otherwise,~$y$ is the $i\\!-\\!(n\\!+\\!1)$-st leaf\n    of~$B$ where~$n$ is the number of nodes of~$A$. Assume that the node~$x$\n    is the $j$-st node of~$T$, then,~$x$ becomes the $j\\!-\\!(n\\!+\\!1)$-st\n    node of~$B$. Hence, the lemma follows by induction hypothesis on~$B$.\n\\end{proof}\n\nThe following proposition is the key of our construction.\n\\begin{Proposition} \\label{prop:FeuillesInversions}\n    Let~$\\sigma$ be a permutation and~$T$ be the left binary search tree\n    obtained by left leaf insertions of the letters of~$\\sigma$, from left\n    to right. Then, the $i\\!+\\!1$-st leaf of~$T$ is right-oriented if and\n    only if~$i$ is a recoil of~$\\sigma$.\n\\end{Proposition}\n\\begin{proof}\n    Set ${\\tt a} := i$ and ${\\tt c} := i\\!+\\!1$. Assume that~${\\tt a}$ is a recoil\n    of~$\\sigma$. We have $\\sigma = u \\, {\\tt c} \\, v \\, {\\tt a} \\, w$ for some\n    words~$u$, $v$, and~$w$. Since no letter~${\\tt b}$ of~$u$ and~$v$ satisfies\n    ${\\tt a} < {\\tt b} < {\\tt c}$, the node of~$T$ labeled by~${\\tt c}$ has a node labeled\n    by~${\\tt a}$ in its left subtree, itself having no right child and thus\n    contributes, by Lemma~\\ref{lem:OrientationFeuille}, to a right-oriented\n    leaf in position~$i\\!+\\!1$.\n\n    Conversely, assume that~${\\tt a}$ is not a recoil of~$\\sigma$. We have\n    $\\sigma = u \\, {\\tt a} \\, v \\, {\\tt c} \\, w$ for some words~$u$, $v$, and~$w$.\n    For the same reason as before, the node of~$T$ labeled by~${\\tt a}$ has\n    a node labeled by~${\\tt c}$ in its right subtree, itself having no left\n    child and thus contributes, by Lemma~\\ref{lem:OrientationFeuille},\n    to a left-oriented leaf in position~$i\\!+\\!1$.\n\\end{proof}\n\nFigure~\\ref{fig:IllustrationOrientationFeuilles} shows an example of\napplication of Proposition~\\ref{prop:FeuillesInversions}.\n\\begin{figure}[ht]\n    \\centering\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0.0,-2){};\n        \\node[Noeud,EtiqClair](1)at(1.0,-1){$1$};\n        \\node[Feuille](2)at(2.0,-4){};\n        \\node[Noeud,EtiqClair](3)at(3.0,-3){$2$};\n        \\node[Feuille](4)at(4.0,-4){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](3)--(4);\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$3$};\n        \\node[Feuille](6)at(6.0,-3){};\n        \\draw[Arete](5)--(3);\n        \\draw[Arete](5)--(6);\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(5);\n        \\node[Noeud,EtiqClair](7)at(7.0,0){$4$};\n        \\node[Feuille](8)at(8.0,-3){};\n        \\node[Noeud,EtiqClair](9)at(9.0,-2){$5$};\n        \\node[Feuille](10)at(10.0,-3){};\n        \\draw[Arete](9)--(8);\n        \\draw[Arete](9)--(10);\n        \\node[Noeud,EtiqClair](11)at(11.0,-1){$6$};\n        \\node[Feuille](12)at(12.0,-3){};\n        \\node[Noeud,EtiqClair](13)at(13.0,-2){$7$};\n        \\node[Feuille](14)at(14.0,-3){};\n        \\draw[Arete](13)--(12);\n        \\draw[Arete](13)--(14);\n        \\draw[Arete](11)--(9);\n        \\draw[Arete](11)--(13);\n        \\draw[Arete](7)--(1);\n        \\draw[Arete](7)--(11);\n    \\end{tikzpicture}}\n    \\caption{The binary search tree drawn with its leaves obtained by left\n    leaf insertions of the letters of $\\sigma := 4136275$, from left to right.\n    The recoils of $\\sigma$ are $2$, $3$, $5$, and $7$ and the $3$-rd, $4$-th,\n    $6$-th, and $8$-th leaves of this binary tree are right-oriented.}\n    \\label{fig:IllustrationOrientationFeuilles}\n\\end{figure}\n\n\\subsubsection{\\texorpdfstring{The ${\\sf P}$-symbol}{The P-symbol}}\n\\begin{Proposition} \\label{prop:PSymboleNonIncr}\n    For any word $u \\in A^*$, the ${\\sf P}$-symbol $(T_L, T_R)$ of~$u$ is\n    a pair of twin binary search trees---$T_L$ (resp.~$T_R$) is a left\n    (resp. right) binary search tree, and the inorder reading of both~$T_L$\n    and~$T_R$ is the nondecreasing rearrangement of~$u$.\n\\end{Proposition}\n\\begin{proof}\n    Note by definition of the {\\sc LeafInsertion} algorithm that~$T_L$\n    (resp.~$T_R$) is a left (resp. right) binary search tree and the inorder\n    reading of both~$T_L$ and~$T_R$ is the nondecreasing rearrangement of~$u$.\n    It is plain that the leaf insertion of~$u$ and~$\\operatorname{std}(u)$ from left to\n    right (resp. right to left) into left (resp. right) binary search trees\n    give binary trees of same shape. That implies that we can consider\n    that $u =: \\sigma$ is a permutation. Proposition~\\ref{prop:FeuillesInversions}\n    implies that the canopies of~$T_L$ and~$T_R$ are complementary because~$i$\n    is a recoil of~$\\sigma$ if and only if~$i$ is not a recoil of~$\\sigma^\\sim$.\n    Thus, the shapes of~$T_L$ and~$T_R$ consist in a pair of twin binary trees.\n\\end{proof}\n\n\\begin{Theoreme} \\label{thm:PSymboleClasses}\n    Let $u, v \\in A^*$. Then, $u {\\:\\equiv_{\\operatorname{B}}\\:} v$ if and only if\n    ${\\sf P}(u) = {\\sf P}(v)$.\n\\end{Theoreme}\n\\begin{proof}\n    Assume $u {\\:\\equiv_{\\operatorname{B}}\\:} v$. Then, by Proposition~\\ref{prop:LienSylv},~$u$\n    and~$v$ are~${\\:\\equiv_{\\operatorname{S}}\\:}$ and ${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$-equivalent. Hence, by\n    Theorem~\\ref{thm:PSymbPBT},~$u$ and~$v$ have the same sylvester and\n    $\\#$-sylvester ${\\sf P}$-symbol, so that ${\\sf P}(u) = {\\sf P}(v)$.\n\n    Conversely assume that ${\\sf P}(u) = {\\sf P}(v) =: (T_L, T_R)$. Since\n    the leaf insertion of both~$u$ and~$v$ from left to right gives~$T_L$,\n    we have, by Theorem~\\ref{thm:PSymbPBT}, $u {\\:\\equiv_{\\operatorname{S}^\\#}\\:} v$. In addition,\n    the leaf insertion of both~$u$ and~$v$ from right to left gives~$T_R$,\n    so that, by the just cited theorem, $u {\\:\\equiv_{\\operatorname{S}}\\:} v$. By\n    Proposition~\\ref{prop:LienSylv}, we have~$u {\\:\\equiv_{\\operatorname{B}}\\:} v$.\n\\end{proof}\n\nIn the case of permutations, each~${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class can be encoded\nby an unlabeled pair of twin binary trees because there is one unique way\nto bijectively label a binary tree with~$n$ nodes on~$\\{1, \\dots, n\\}$\nsuch that it is a binary search tree. Hence, in the sequel, unlabeled pairs\nof twin binary search trees can be considered as labeled by a permutation,\nand conversely.\n\n\\subsubsection{\\texorpdfstring{The ${\\sf Q}$-symbol}{The Q-symbol}}\nLet us recall the following lemma of~\\cite{HNT05}, restated in our setting\nand supplemented with a respective part:\n\\begin{Lemme} \\label{lem:FormeInsIncr}\n    Let~$u$ be a word and $\\sigma := \\operatorname{std}(u)^{-1}$. The right (resp. left)\n    binary search tree obtained by inserting $u$ from right to left (resp.\n    from left to right) and~$\\operatorname{decr}(\\sigma)$ (resp.~$\\operatorname{incr}(\\sigma)$) have\n    same shape.\n\\end{Lemme}\n\n\\begin{Proposition} \\label{prop:TypeQSymbole}\n    For any word $u \\in A^*$, the shape of the ${\\sf Q}$-symbol~$(S_L, S_R)$\n    of~$u$ is a pair of twin binary trees. Moreover,~$S_L$ is an increasing\n    binary tree,~$S_R$ is a decreasing binary tree and their inorder reading\n    is~$\\operatorname{std}(u)^{-1}$.\n\\end{Proposition}\n\\begin{proof}\n    By definition of the~${\\sf Q}$-symbol,~$S_L$ and~$S_R$ are respectively\n    the increasing and the decreasing binary trees of $\\sigma := \\operatorname{std}(u)^{-1}$.\n    By Lemma~\\ref{lem:FormeInsIncr}, a binary tree with same shape as~$S_L$\n    (resp.~$S_R$) can also be obtained by leaf insertions of the letters\n    of~$\\sigma^{-1}$ from left to right (resp. right to left). Thus, by\n    Proposition~\\ref{prop:FeuillesInversions}, the shape of $(S_L, S_R)$ is a pair\n    of twin binary trees. Moreover, by the definition of the algorithms~$\\operatorname{incr}$\n    and~$\\operatorname{decr}$, we can prove by induction on the size of~$\\sigma$ that\n    the binary trees~$S_L$ and~$S_R$ have both~$\\sigma$ as inorder reading.\n\\end{proof}\n\n\\begin{Theoreme} \\label{thm:BijectionMotsPQSymbole}\n    The map $u \\longmapsto \\left({\\sf P}(u), {\\sf Q}(u)\\right)$ is a bijection\n    between the elements of $A^*$ and the set formed by the pairs\n    $\\left((T_L, T_R), (S_L, S_R)\\right)$ where\n    \\begin{enumerate}[label = (\\roman*)]\n        \\item $(T_L, T_R)$ is a pair of twin binary search trees---$T_L$\n        (resp. $T_R$) is a left (resp. right) binary search tree, and $T_L$\n        and $T_R$ have both the same inorder reading; \\label{item:BMPQS1}\n        \\item $(S_L, S_R)$ is a pair of twin binary trees where $S_L$ (resp. $S_R$)\n        is an increasing (resp. decreasing) binary tree, and $S_L$ and $S_R$\n        have both the same inorder reading; \\label{item:BMPQS2}\n        \\item $(T_L, T_R)$ and $(S_L, S_R)$ have same shape. \\label{item:BMPQS3}\n    \\end{enumerate}\n\\end{Theoreme}\n\\begin{proof}\n    Let us first show that for any $u \\in A^*$, the pair\n    $\\left({\\sf P}(u), {\\sf Q}(u)\\right)$ satisfies the assertions of the\n    theorem. Point~\\ref{item:BMPQS1} follows from\n    Proposition~\\ref{prop:PSymboleNonIncr}. Point~\\ref{item:BMPQS2} follows\n    from Proposition~\\ref{prop:TypeQSymbole}. Moreover, by Lemma~\\ref{lem:FormeInsIncr},\n    Point~\\ref{item:BMPQS3} checks out. Besides, as already mentioned,\n    it is possible to reconstruct from the pair $\\left({\\sf P}(u), {\\sf Q}(u)\\right)$\n    the word~$u$ and such a word is unique. That shows that the correspondence\n    is well-defined and injective.\n\n    Conversely, assume that $\\left((T_L, T_R), (S_L, S_R)\\right)$ satisfies\n    the three assertions of the theorem. According to~\\cite{HNT02}, there\n    is a bijection between the elements of~$A^*$ and the pairs $(T_R, S_R)$\n    where~$T_R$ is a right binary search tree and~$S_R$ a decreasing binary\n    tree of same shape. Let~$u$ be the word in correspondence with $(T_R, S_R)$.\n    In the same way, there is a bijection between the elements of~$A^*$\n    and the pairs $(T_L, S_L)$ where~$T_L$ is a left binary search tree\n    and~$S_L$ an increasing binary tree of same shape. Let~$v$ be the word\n    in correspondence with $(T_L, S_L)$. By hypothesis,~$T_L$ and~$T_R$\n    have both the same inorder reading, implying $\\operatorname{ev}(u) = \\operatorname{ev}(v)$.\n    In the same way, since~$S_L$ and~$S_R$ have both the same inorder reading,\n    one has $\\operatorname{std}(u)^{-1} = \\operatorname{std}(v)^{-1}$. Hence, we have $\\operatorname{std}(u) = \\operatorname{std}(v)$\n    and thus~$u = v$. Note also that the pair $(T_L, S_L)$ is entirely\n    determined by the pair $(T_R, S_R)$ and conversely. Now, again according\n    to~\\cite{HNT02}, the pair $(T_R, S_R)$ is the sylvester ${\\sf P}$-symbol\n    of~$u$ and the pair $(T_L, S_R)$ is the $\\#$-sylvester ${\\sf P}$-symbol\n    of~$u$. Hence, the insertion of $u$ gives the pair\n    $\\left((T_L, T_R), (S_L, S_R)\\right)$, showing that the correspondence\n    is also surjective.\n\\end{proof}\n\n\\subsection{Distinguished permutations from a pair of twin binary trees}\nWe present in this section some algorithms to read some distinguished\npermutations from a pair of twin binary search trees. Let us first start\nwith a useful characterization of ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes.\n\n\\subsubsection{Baxter equivalence classes as linear extensions of posets}\nLet $T$ be an $A$-labeled binary tree. We shall denote by $\\bigtriangleup(T)$\n(resp. $\\bigtriangledown(T)$) the poset $(N, \\leq)$ where $N := \\{1, \\dots, n\\}$,\n$n$ is the number of nodes of~$T$, and $\\leq$ is defined, for $i, j \\in N$, by\n\\begin{equation}\n    i \\leq j \\qquad\n    \\mbox{if the $i$-th node is an ancestor (resp. descendant) of the\n    $j$-th node of~$T$}.\n\\end{equation}\nIf the sequence~$i_1 \\dots i_n$ is a linear extension of~$\\bigtriangleup(T)$\n(resp.~$\\bigtriangledown(T)$), we shall also say that the word~$u_1 \\dots u_n$ is\na \\emph{linear extension} of~$\\bigtriangleup(T)$ (resp.~$\\bigtriangledown(T)$) if for any\n$1 \\leq \\ell \\leq n$, the label of the $i_\\ell$-th node of~$T$ is~$u_\\ell$.\n\\medskip\n\nThe words of a sylvester equivalence class encoded by a labeled right\nbinary search tree~$T$ coincide with the linear extensions of~$\\bigtriangledown(T)$\n(see Note 4 of~\\cite{HNT05}). Additionally, this also says that the words\nof a $\\#$-sylvester equivalence class encoded by a labeled left binary\nsearch tree~$T$ are exactly the linear extensions of~$\\bigtriangleup(T)$. One has\na similar characterization of Baxter equivalence classes:\n\\begin{Proposition} \\label{prop:BXExtLin}\n    The words of a Baxter equivalence class encoded by a pair of twin binary\n    search trees~$(T_L, T_R)$ coincide with the words that are both\n    linear extensions of the posets~$\\bigtriangleup(T_L)$ and~$\\bigtriangledown(T_R)$.\n\\end{Proposition}\n\\begin{proof}\n    Let~$u$ be a word belonging to the Baxter equivalence class encoded\n    by~$(T_L, T_R)$. By Theorem~\\ref{thm:PSymboleClasses},~$T_L$ (resp.~$T_R$)\n    can be obtained by leaf inserting~$u$ from left to right (resp. right\n    to left). Hence, if $i \\leq j$  in~$\\bigtriangleup(T_L)$ (resp. in~$\\bigtriangledown(T_R)$)\n    then~$i$ is smaller than~$j$ as integers. Thus,~$u$ is a linear extension\n    of both~$\\bigtriangleup(T_L)$ and~$\\bigtriangledown(T_R)$.\n\n    Assume now that~$u$ is a linear extension of~$\\bigtriangleup(T_L)$ and~$\\bigtriangledown(T_R)$\n    and let $v$ be any word of the Baxter equivalence class encoded by $(T_L, T_R)$.\n    By Theorem~\\ref{thm:PSymboleClasses},~$T_L$ (resp.~$T_R$) can be obtained\n    by leaf inserting~$v$ from left to right (resp. right to left).\n    Note 4 of~\\cite{HNT05} implies that~$u {\\:\\equiv_{\\operatorname{S}^\\#}\\:} v$ and~$u {\\:\\equiv_{\\operatorname{S}}\\:} v$.\n    Hence, by Proposition~\\ref{prop:LienSylv}, one has~$u {\\:\\equiv_{\\operatorname{B}}\\:} v$, showing\n    that~$u$ also belongs to the Baxter equivalence class represented by~$(T_L, T_R)$.\n\\end{proof}\n\nTo illustrate Proposition~\\ref{prop:BXExtLin}, consider the following\nlabeled pair of twin binary search trees,\n\\begin{equation}\n    \\begin{split} (T_L, T_R) := \\: \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n        \\node[Noeud,EtiqClair](1)at(1,-1){$2$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2,-3){$3$};\n        \\node[Noeud,EtiqClair](3)at(3,-2){$4$};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud,EtiqClair](4)at(4,0){$5$};\n        \\draw[Arete](4)--(1);\n        \\node[Noeud,EtiqClair](5)at(5,-2){$6$};\n        \\node[Noeud,EtiqClair](6)at(6,-1){$7$};\n        \\draw[Arete](6)--(5);\n        \\draw[Arete](4)--(6);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n        \\node[Noeud,EtiqClair](1)at(1,-3){$2$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqClair](2)at(2,-1){$3$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqClair](3)at(3,-2){$4$};\n        \\node[Noeud,EtiqClair](4)at(4,-3){$5$};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](2)--(3);\n        \\node[Noeud,EtiqClair](5)at(5,0){$6$};\n        \\draw[Arete](5)--(2);\n        \\node[Noeud,EtiqClair](6)at(6,-1){$7$};\n        \\draw[Arete](5)--(6);\n    \\end{tikzpicture}}\n    \\end{split}\\,.\n\\end{equation}\nThe set of words that are linear extensions of~$\\bigtriangleup(T_L)$\nand~$\\bigtriangledown(T_R)$ are (the highlighted permutation is a Baxter permutation)\n\\begin{align}\n    \\begin{split}\n        \\{{\\bf 5214376}, & \\enspace 5214736, \\enspace\n            5217436, \\enspace 5241376, \\enspace 5241736, \\\\\n            5247136, & \\enspace 5271436, \\enspace\n            5274136, \\enspace 5721436, \\enspace 5724136 \\},\n    \\end{split}\n\\end{align}\nwhich is exactly the Baxter equivalence class encoded by~$(T_L, T_R)$.\n\\medskip\n\nNote that it is possible to represent the order relations induced by the\nposets~$\\bigtriangleup(T_L)$ and $\\bigtriangledown(T_R)$ in only one poset\n$\\bigtriangleup(T_L) \\cup \\bigtriangledown(T_R)$, adding  on~$\\bigtriangleup(T_L)$ the order\nrelations induced by~$\\bigtriangledown(T_R)$. For the previous example, we obtain\nthe poset\n\\begin{equation}\n    \\begin{split}\\bigtriangleup(T_L) \\cup \\bigtriangledown(T_R) = \\:\\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n        \\node[Noeud,EtiqClair](1)at(1,-1){$2$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2,-3){$3$};\n        \\node[Noeud,EtiqClair](3)at(3,-2){$4$};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud,EtiqClair](4)at(4,0){$5$};\n        \\draw[Arete](4)--(1);\n        \\node[Noeud,EtiqClair](5)at(5,-4){$6$};\n        \\node[Noeud,EtiqClair](6)at(6,-1){$7$};\n        \\draw[Arete](6)--(5);\n        \\draw[Arete](4)--(6);\n        \\draw[Arete](0)--(2);\n        \\draw[Arete](2)--(5);\n    \\end{tikzpicture}}\n    \\end{split}\\,.\n\\end{equation}\n\n\\subsubsection{Extracting Baxter permutations}\nThe following algorithm allows, given an $A$-labeled pair of twin binary\nsearch trees~$(T_L, T_R)$, to compute a word belonging to the\n${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by~$(T_L, T_R)$. When~$(T_L, T_R)$\nis labeled by a permutation, our algorithm coincides with the algorithm\ndesigned by Dulucq and Guibert to describe a bijection between pairs of\ntwin binary trees and Baxter permutations~\\cite{DG94}. Besides, since their\nalgorithm always computes a Baxter permutation, our algorithm also returns\na Baxter permutation when~$(T_L, T_R)$ is labeled by a permutation.\n\\medskip\n\n{\\flushleft\n    {\\bf Algorithm:} {\\sc ExtractBaxter}. \\\\\n    {\\bf Input:} An $A$-labeled pair of twin binary search trees~$(T_L, T_R)$. \\\\\n    {\\bf Output:} A word belonging to the Baxter equivalence class encoded\n    by~$(T_L, T_R)$. \\\\\n    \\begin{enumerate}\n        \\item Let $u := \\epsilon$ be the empty word.\n        \\item While $T_L \\ne \\perp$ and $T_R \\ne \\perp$:\n        \\begin{enumerate}\n            \\item Let ${\\tt a}$ be the label of the root of $T_L$.\n            \\item Let $i$ be the index of root of $T_L$.\n            \\item Set $u := u{\\tt a}$.\n            \\item Let $A$ (resp. $B$) be the left (resp. right) subtree of $T_L$.\n            \\item If the $i$-th node of $T_R$ is a left child in $T_R$:\n            \\begin{enumerate}\n                \\item Then, set $T_L := A {\\,\\diagup\\,} B$.\n                \\item Otherwise, set $T_L := A {\\,\\diagdown\\,} B$.\n            \\end{enumerate}\n            \\item Suppress the $i$-th node in $T_R$.\n        \\end{enumerate}\n        \\item Return $u$.\n    \\end{enumerate}\n    {\\bf End.}\n}\n\\medskip\n\nFigure~\\ref{fig:ExempleExtractBaxter} shows an execution of this algorithm.\n\\begin{figure}[ht]\n    \\centering\n    \\begin{equation*}\n        \\begin{split} (T_L, T_R) := \\: \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$2$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,-2){$3$};\n            \\node[Noeud,EtiqClair](3)at(3,-3){$4$};\n            \\draw[Arete](2)--(3);\n            \\draw[Arete](1)--(2);\n            \\node[Noeud,EtiqFonce](4)at(4,0){$5$};\n            \\draw[Arete](4)--(1);\n            \\node[Noeud,EtiqClair](5)at(5,-1){$6$};\n            \\draw[Arete](4)--(5);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\enspace\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-3){$2$};\n            \\draw[Arete](0)--(1);\n            \\node[Noeud,EtiqClair](2)at(2,-1){$3$};\n            \\draw[Arete](2)--(0);\n            \\node[Noeud,EtiqClair](3)at(3,0){$4$};\n            \\draw[Arete](3)--(2);\n            \\node[Noeud,EtiqFonce](4)at(4,-2){$5$};\n            \\node[Noeud,EtiqClair](5)at(5,-1){$6$};\n            \\draw[Arete](5)--(4);\n            \\draw[Arete](3)--(5);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split} \\quad \\xrightarrow{{\\tt a} = 5} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$2$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,-2){$3$};\n            \\node[Noeud,EtiqClair](3)at(3,-3){$4$};\n            \\draw[Arete](2)--(3);\n            \\draw[Arete](1)--(2);\n            \\node[Noeud,EtiqFonce](4)at(4,0){$6$};\n            \\draw[Arete](4)--(1);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\enspace\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-3){$2$};\n            \\draw[Arete](0)--(1);\n            \\node[Noeud,EtiqClair](2)at(2,-1){$3$};\n            \\draw[Arete](2)--(0);\n            \\node[Noeud,EtiqClair](3)at(3,0){$4$};\n            \\draw[Arete](3)--(2);\n            \\node[Noeud,EtiqFonce](4)at(4,-1){$6$};\n            \\draw[Arete](3)--(4);\n        \\end{tikzpicture}}\n        \\end{split}\n    \\end{equation*}\n    \\begin{equation*}\n        \\begin{split}\\xrightarrow{{\\tt a} = 6} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-1){$1$};\n            \\node[Noeud,EtiqFonce](1)at(1,0){$2$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,-1){$3$};\n            \\node[Noeud,EtiqClair](3)at(3,-2){$4$};\n            \\draw[Arete](2)--(3);\n            \\draw[Arete](1)--(2);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\enspace\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqFonce](1)at(1,-3){$2$};\n            \\draw[Arete](0)--(1);\n            \\node[Noeud,EtiqClair](2)at(2,-1){$3$};\n            \\draw[Arete](2)--(0);\n            \\node[Noeud,EtiqClair](3)at(3,0){$4$};\n            \\draw[Arete](3)--(2);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\\quad \\xrightarrow{{\\tt a} = 2} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$3$};\n            \\node[Noeud,EtiqClair](2)at(2,-2){$4$};\n            \\draw[Arete](1)--(2);\n            \\draw[Arete](0)--(1);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\enspace\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$3$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,0){$4$};\n            \\draw[Arete](2)--(1);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\\quad \\xrightarrow{{\\tt a} = 1} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$3$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$4$};\n            \\draw[Arete](0)--(1);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\enspace\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,-1){$3$};\n            \\node[Noeud,EtiqClair](1)at(1,0){$4$};\n            \\draw[Arete](1)--(0);\n        \\end{tikzpicture}}\n        \\end{split}\n    \\end{equation*}\n    \\begin{equation*}\n        \\begin{split} \\xrightarrow{{\\tt a} = 3} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{%\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$4$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\enspace\n        \\begin{split}\n        \\scalebox{.35}{%\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$4$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\\quad \\xrightarrow{{\\tt a} = 5} \\quad \\end{split}\n        \\begin{split} \\perp \\perp \\end{split}\n    \\end{equation*}\n    \\caption{An execution of the algorithm {\\sc ExtractBaxter} on~$(T_L, T_R)$.\n    The computed Baxter permutation is~$562134$.}\n    \\label{fig:ExempleExtractBaxter}\n\\end{figure}\n\\medskip\n\nThe results of Dulucq and Guibert~\\cite{DG94} imply that {\\sc ExtractBaxter}\nterminates. The only thing to prove is that the computed word belongs to\nthe ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by the pair of twin binary search\ntrees as input. For that, let us first prove the following lemma.\n\n\\begin{Lemme} \\label{lem:NoeudRacineFeuille}\n    Let~$(T_L, T_R)$ be a non-empty pair of twin binary trees. If the root\n    of~$T_L$ is the $i$-th node of~$T_L$, then, the $i$-th node of~$T_R$\n    has no child.\n\\end{Lemme}\n\\begin{proof}\n    Assume that $T_L = A \\wedge B$. Note that if both~$A$ and~$B$ are\n    empty,~$T_L$ and~$T_R$ are the one-node binary trees and the lemma is\n    clearly satisfied.\n\n    If $A \\ne \\perp$, assume that the $i$-th node of~$T_R$ has a non-empty\n    left subtree. That implies that the $i$-th leaf of~$T_R$ is not attached\n    to its $i$-th node. Thus, by Lemma~\\ref{lem:OrientationFeuille}, the $i$-th\n    leaf of~$T_R$ is attached to its $i\\!-\\!1$-st node and is right-oriented.\n    In~$T_L$, the $i$-th leaf cannot be attached to its $i$-th node\n    because~$A \\ne \\perp$. Hence, by Lemma~\\ref{lem:OrientationFeuille},\n    the $i$-th leaf of~$T_L$ is also attached to its $i\\!-\\!1$-st node and is\n    right-oriented. Since~$T$ contains at least~$i$ nodes, there is at\n    least~$i\\!+\\!1$ leaves in~$T$, implying that the $i$-th leaf is not\n    the rightmost leaf of~$T_L$ and~$T_R$, and thus~$(T_L, T_R)$ is not\n    a pair of twin binary trees, contradicting the hypothesis.\n\n    Assume now that the $i$-th node of~$T_R$ has a non-empty right subtree.\n    That implies that the $i\\!+\\!1$-st leaf of~$T_R$ is not attached to\n    its $i$-th node and thus, by Lemma~\\ref{lem:OrientationFeuille},\n    the $i\\!+\\!1$-st leaf of~$T_R$ is left-oriented. Moreover, since the\n    $i$-th node of~$T_R$ has a non-empty right subtree and the $i$-th node\n    of~$T_L$ is its root, the $i$-th node of~$T_L$ also has a non-empty\n    right subtree. That implies that the $i\\!+\\!1$-st leaf of~$T_L$ is not\n    attached to its $i$-th node and thus, by Lemma~\\ref{lem:OrientationFeuille},\n    the $i\\!+\\!1$-st leaf of~$T_R$ is also left-oriented. That contradicts\n    that~$(T_L, T_R)$ is a pair of twin binary trees, and implies that the\n    $i$-th node of~$T_R$ has no child. The case~$B \\ne \\perp$ is analogous.\n\\end{proof}\n\n\\begin{Proposition} \\label{prop:LectureBaxMemeClasse}\n    For any $A$-labeled pair of twin binary search trees~$(T_L, T_R)$ as\n    input, the algorithm {\\sc ExtractBaxter} computes a word belonging to\n    the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by~$(T_L, T_R)$. Moreover,\n    if~$(T_L, T_R)$ is labeled by a permutation, the computed word is a\n    Baxter permutation.\n\\end{Proposition}\n\\begin{proof}\n    Let us prove by induction on~$n$, that is the number of nodes of~$T_L$\n    and~$T_R$, that if~$(T_L, T_R)$ is an $A$-labeled pair of twin binary\n    search trees, then {\\sc ExtractBaxter} returns a word that is a linear\n    extension of~$\\bigtriangleup(T_L)$ and a linear extension of~$\\bigtriangledown(T_R)$,\n    \\emph{i.e.}, by Proposition~\\ref{prop:BXExtLin}, a word belonging to\n    the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by~$(T_L, T_R)$. This property\n    clearly holds for~$n \\leq 1$. Now, assume that $T_L = A \\wedge_{\\tt a} B$\n    where~${\\tt a}$ is the label of the root of~$T_L$. By\n    Lemma~\\ref{lem:NoeudRacineFeuille}, if the root of~$T_L$ is its $i$-th\n    node, the $i$-th node~$x$ of~$T_R$ has no child. Moreover, since~$T_L$\n    and~$T_R$ are binary search trees and labeled by a same word, their\n    respective $i$-th nodes have the same label~${\\tt a}$. Moreover, the canopy\n    of~$T_L$ is of the form~$v01w$ where~$v := \\operatorname{cnp}(A)$ and~$w := \\operatorname{cnp}(B)$,\n    and the canopy of~$T_R$ is of the form~$v'10w'$ where~$v'$ (resp.~$w'$)\n    is the complementary of~$v$ (resp.~$w$) since that~$(T_L, T_R)$ is a\n    pair of twin binary trees. We have now two cases whether~$x$ is a left\n    of right child in~$T_R$.\n\n    If~$x$ is a left child in~$T_R$, the algorithm returns the word~${\\tt a} u$\n    where~$u$ is the word obtained by applying the algorithm on~$(T'_L, T'_R)$\n    where~$T'_L = A {\\,\\diagup\\,} B$ and~$T'_R$ is obtained from~$T_R$ by suppressing\n    the node~$x$. First, the canopy of~$T'_L$ is of the form~$v0w$ and the\n    canopy of~$T'_R$ is of the form~$v'1w'$. Moreover,~$T'_L$ and~$T'_R$\n    are clearly still binary search trees. That implies that~$(T'_L, T'_R)$\n    is a pair of twin binary search trees. By induction hypothesis and\n    Proposition~\\ref{prop:BXExtLin}, the word~$u$ belongs to the\n    ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by~$(T'_L, T'_R)$, and thus,~${\\tt a} u$\n    belongs to the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by~$(T_L, T_R)$\n    because~${\\tt a} u$ is a linear extension of~$\\bigtriangleup(T_L)$ (resp.~$\\bigtriangledown(T_R)$)\n    since~$u$ is a linear extension of both~$\\bigtriangleup(T'_L)$ and~$\\bigtriangledown(T'_R)$.\n    The case where~$x$ is a right child in~$T_R$ is analogous.\n\n    Finally, when~$(T_L, T_R)$ is labeled by a permutation, {\\sc ExtractBaxter}\n    coincides with the algorithm of Dulucq and Guibert~\\cite{DG94} and\n    computes a Baxter permutation.\n\\end{proof}\n\nThe validity of {\\sc ExtractBaxter} implies the two following results.\n\n\\begin{Theoreme} \\label{thm:BijectionEquivBXPermuBX}\n    For any~$n \\geq 0$, there is a bijection between the set of Baxter\n    equivalence classes of words of length $n$ and $A$-labeled pairs of\n    twin binary search trees with~$n$ nodes.\n\\end{Theoreme}\n\\begin{proof}\n    By Proposition~\\ref{prop:PSymboleNonIncr} and Theorem~\\ref{thm:PSymboleClasses},\n    the ${\\sf P}$-symbol algorithm induces an injection between the set\n    of equivalence classes of~$\\mathfrak{S}_n \/_{\\:\\equiv_{\\operatorname{B}}\\:}$ and the set of\n    unlabeled pairs of twin binary trees. Moreover,\n    by Proposition~\\ref{prop:LectureBaxMemeClasse}, the algorithm\n    {\\sc ExtractBaxter} exhibits a surjection between these two sets.\n    Hence, these two sets are in bijection.\n\\end{proof}\n\nTheorem~\\ref{thm:BijectionEquivBXPermuBX} implies in particular that\nthe Baxter equivalence classes of permutations of size~$n$ are in bijection\nwith pairs of twin binary trees labeled by a permutation (or equivalently with\nunlabeled pairs of twin binary trees).\n\n\\begin{Theoreme} \\label{thm:EquivBXBaxter}\n    For any~$n \\geq 0$, each equivalence class of~$\\mathfrak{S}_n \/_{\\:\\equiv_{\\operatorname{B}}\\:}$\n    contains exactly one Baxter permutation.\n\\end{Theoreme}\n\\begin{proof}\n    Let~$C$ be an equivalence class of~$\\mathfrak{S}_n \/_{\\:\\equiv_{\\operatorname{B}}\\:}$. By\n    Theorem~\\ref{thm:BijectionEquivBXPermuBX},~$C$ can be represented by\n    an unlabeled pair of twin binary trees~$J$. By\n    Proposition~\\ref{prop:LectureBaxMemeClasse}, the algorithm\n    {\\sc ExtractBaxter} computes a permutation belonging to the\n    ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by~$J$, showing that each\n    ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class of permutations contains at least one\n    Baxter permutation. The theorem follows from the fact that Baxter\n    permutations are equinumerous with unlabeled pairs of twin binary trees.\n\\end{proof}\n\n\\subsubsection{Extracting minimal and maximal permutations}\nReading defined in~\\cite{Rea05} \\emph{twisted Baxter permutations}, that\nare the permutations avoiding the generalized permutation patterns~$2-41-3$\nand~$3-41-2$. These permutations are particular elements of Baxter classes\nof permutations:\n\\begin{Proposition} \\label{prop:TwistedMinClasses}\n    Twisted Baxter permutations coincide with minimal elements of\n    Baxter equivalence classes of permutations.\n\\end{Proposition}\n\\begin{proof}\n    First, note that by Proposition~\\ref{prop:EquivBXInter}, every Baxter\n    equivalence class of permutations has a minimal element. Assume that~$\\sigma$\n    is minimal of its ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class of permutations. Then,\n    it is not possible to perform any rewriting of the form\n    \\begin{equation}\n        {\\tt b} \\, u \\, {\\tt d} {\\tt a} \\, v \\, {\\tt b}' \\rightarrow\n        {\\tt b} \\, u \\, {\\tt a} {\\tt d} \\, v \\, {\\tt b}',\n    \\end{equation}\n    where ${\\tt a} < {\\tt b}, {\\tt b}' < {\\tt d}$ are letters, and~$u$ and~$v$ are words.\n    Hence,~$\\sigma$ avoids the patterns $2-41-3$ and $3-41-2$, and is a\n    twisted Baxter permutation.\n\n    Conversely, if~$\\sigma$ is a twisted Baxter permutation, it avoids\n    $2-41-3$ and $3-41-2$ and it is not possible to perform any\n    rewriting~$\\rightarrow$, so that, by Proposition~\\ref{prop:EquivBXInter}\n    and Lemma~\\ref{lem:DiagHasseEqBX}, it is minimal of its\n    ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class.\n\\end{proof}\n\nIn a similar way, by calling \\emph{anti-twisted Baxter permutation} any\npermutation that avoids the generalized permutation patterns $2-14-3$ and\n$3-14-2$, an analogous proof to the one of Proposition~\\ref{prop:TwistedMinClasses}\nshows that anti-twisted Baxter permutations coincide with maximal\nelements of Baxter equivalence classes of permutations.\n\\medskip\n\nProposition~\\ref{prop:TwistedMinClasses} implies that twisted Baxter\npermutations, anti-twisted Baxter permutations, and Baxter permutations\nare equinumerous since by Theorem~\\ref{thm:EquivBXBaxter} there is exactly\none Baxter permutation by ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class of permutations and\nby Proposition~\\ref{prop:EquivBXInter}, there is also exactly one twisted\n(and one anti-twisted) Baxter permutation. This suggests among other that\nthere exists a bijection sending a Baxter permutation to the twisted Baxter\npermutation of its ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class.\n\\medskip\n\nAs pointed out by Law and Reading, West has shown first a bijection between\nBaxter permutations and twisted Baxter permutations using generating\ntrees~\\cite{B03}. In our setting, as in the setting of Law and Reading~\\cite{LR10},\nthis bijection is the one preserving the classes. Here follows an algorithm\nto compute this bijection.\n\\medskip\n\nLet us consider the following algorithm which allows, given an $A$-labeled\npair of twin binary search trees~$(T_L, T_R)$, to compute the minimal\npermutation for the lexicographic order belonging to the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\nclass encoded by~$(T_L, T_R)$.\n\\medskip\n\n{\\flushleft\n    {\\bf Algorithm:} {\\sc ExtractMin}. \\\\\n    {\\bf Input:} An $A$-labeled pair of twin binary search trees~$(T_L, T_R)$. \\\\\n    {\\bf Output:} The minimal word for the lexicographic order of the\n    class encoded by~$(T_L, T_R)$. \\\\\n    \\begin{enumerate}\n        \\item Let $u := \\epsilon$ be the empty word.\n        \\item Let $F := T_L$ be a rooted forest.\n        \\item While $F$ is not empty and $T_R \\ne \\perp$:\n        \\begin{enumerate}\n            \\item Let $i$ be the smallest index such that the $i$-th node\n            of $F$ is a root and the $i$-th node of $T_R$ has no child.\\label{item:InstrChoixNoeudMin}\n            \\item Let ${\\tt a}$ be the label of the $i$-th node of $T_L$.\n            \\item Set $u := u {\\tt a}$.\n            \\item Suppress the $i$-th node of $F$ and the $i$-th node of $T_R$.\n        \\end{enumerate}\n        \\item Return $u$.\n    \\end{enumerate}\n    {\\bf End.}\n}\n\\medskip\n\nNote that, by choosing in the instruction~(\\ref{item:InstrChoixNoeudMin}) the\ngreatest index instead of the smallest, the previous algorithm would compute the\nmaximal word for the lexicographic order of the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class\nencoded by~$(T_L, T_R)$. Let us call this variant {\\sc ExtractMax}.\n\\medskip\n\nFigure~\\ref{fig:ExempleExtractMin} shows an example of application of {\\sc ExtractMin}.\n\\begin{figure}[ht]\n    \\centering\n    \\begin{equation*}\n        \\begin{split}(T_L, T_R) := \\: \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$2$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,-2){$3$};\n            \\node[Noeud,EtiqClair](3)at(3,-3){$4$};\n            \\draw[Arete](2)--(3);\n            \\draw[Arete](1)--(2);\n            \\node[Noeud,EtiqFonce](4)at(4,0){$5$};\n            \\draw[Arete](4)--(1);\n            \\node[Noeud,EtiqClair](5)at(5,-1){$6$};\n            \\draw[Arete](4)--(5);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\quad\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-3){$2$};\n            \\draw[Arete](0)--(1);\n            \\node[Noeud,EtiqClair](2)at(2,-1){$3$};\n            \\draw[Arete](2)--(0);\n            \\node[Noeud,EtiqClair](3)at(3,0){$4$};\n            \\draw[Arete](3)--(2);\n            \\node[Noeud,EtiqFonce](4)at(4,-2){$5$};\n            \\node[Noeud,EtiqClair](5)at(5,-1){$6$};\n            \\draw[Arete](5)--(4);\n            \\draw[Arete](3)--(5);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\\quad \\xrightarrow{{\\tt a} = 5} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-1){$1$};\n            \\node[Noeud,EtiqFonce](1)at(1,0){$2$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,-1){$3$};\n            \\node[Noeud,EtiqClair](3)at(3,-2){$4$};\n            \\draw[Arete](2)--(3);\n            \\draw[Arete](1)--(2);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,0){$6$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\quad\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqFonce](1)at(1,-3){$2$};\n            \\draw[Arete](0)--(1);\n            \\node[Noeud,EtiqClair](2)at(2,-1){$3$};\n            \\draw[Arete](2)--(0);\n            \\node[Noeud,EtiqClair](3)at(3,0){$4$};\n            \\draw[Arete](3)--(2);\n            \\node[Noeud,EtiqClair](4)at(4,-1){$6$};\n            \\draw[Arete](3)--(4);\n        \\end{tikzpicture}}\n        \\end{split}\n    \\end{equation*}\n    \\begin{equation*}\n        \\begin{split}\\xrightarrow{{\\tt a} = 2} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{%\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$1$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,0){$3$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$4$};\n            \\draw[Arete](0)--(1);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,0){$6$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\quad\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$3$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,0){$4$};\n            \\draw[Arete](2)--(1);\n            \\node[Noeud,EtiqClair](3)at(3,-1){$6$};\n            \\draw[Arete](2)--(3);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\\quad \\xrightarrow{{\\tt a} = 1} \\quad\\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$3$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$4$};\n            \\draw[Arete](0)--(1);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{%\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,0){$6$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\quad\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,-1){$3$};\n            \\node[Noeud,EtiqClair](1)at(1,0){$4$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,-1){$6$};\n            \\draw[Arete](1)--(2);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\\quad \\xrightarrow{{\\tt a} = 3} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{%\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,0){$4$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{%\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$6$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\quad\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,0){$4$};\n            \\node[Noeud,EtiqFonce](1)at(1,-1){$6$};\n            \\draw[Arete](0)--(1);\n        \\end{tikzpicture}}\n        \\end{split}\n    \\end{equation*}\n    \\begin{equation*}\n        \\begin{split}\\xrightarrow{{\\tt a} = 6} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{%\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$4$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\quad\n        \\begin{split}\n        \\scalebox{.35}{%\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$4$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\\quad \\xrightarrow{{\\tt a} = 4} \\quad \\end{split}\n        \\begin{split} \\perp \\perp \\end{split}\n    \\end{equation*}\n    \\caption{An execution of the algorithm {\\sc ExtractMin} on~$(T_L, T_R)$.\n    The computed permutation is~$521364$ and it is minimal in\n    its~${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class.}\n    \\label{fig:ExempleExtractMin}\n\\end{figure}\n\n\\begin{Proposition} \\label{prop:LectureMin}\n    For any $A$-labeled pair of twin binary search trees~$(T_L, T_R)$\n    as input, the algorithm {\\sc ExtractMin} (resp. {\\sc ExtractMax})\n    computes the minimal (resp. maximal) word for the lexicographic order\n    of the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by~$(T_L, T_R)$. Moreover,\n    if~$(T_L, T_R)$ is labeled by a permutation, the computed word is the\n    minimal (resp. maximal) permutation for the permutohedron order of its\n    ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class.\n\\end{Proposition}\n\\begin{proof}\n    The output~$u$ of the algorithm {\\sc ExtractMin} (resp. {\\sc ExtractMax})\n    is both a linear extension of~$\\bigtriangleup(T_L)$ and a linear extension\n    of~$\\bigtriangledown(T_R)$. That implies by Proposition~\\ref{prop:BXExtLin}\n    that~$u$ belongs to the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by\n    the input pair of twin binary trees. Moreover, this algorithm terminates\n    since by Theorem~\\ref{thm:EquivBXBaxter}, each $A$-labeled pair of\n    twin binary search trees~$(T_L, T_R)$ admits at least one word that\n    is a common linear extension of~$\\bigtriangleup(T_L)$ and~$\\bigtriangledown(T_R)$.\n    The minimality (resp. maximality) for the lexicographic order of the\n    computed word comes from the fact that at each step, the node that\n    has the smallest (resp. greatest) label is chosen.\n\n    Finally, since the lexicographic order is a linear extension of the\n    permutohedron order, and by Proposition~\\ref{prop:EquivBXInter}, since\n    Baxter equivalence classes are intervals of the permutohedron,\n    {\\sc ExtractMin} (resp. {\\sc ExtractMax}) returns the minimal (resp.\n    maximal) permutation for the permutohedron order of its Baxter\n    equivalence class.\n\\end{proof}\n\nBy Proposition~\\ref{prop:LectureMin} and using our Robinson-Schensted-like\nalgorithm, we can compute the bijection between Baxter permutations and\ntwisted Baxter permutations in the following way: If~$\\sigma$ is a Baxter\npermutation, apply {\\sc ExtractMin} on~${\\sf P}(\\sigma)$ to obtain its\ncorresponding twisted Baxter permutation. Conversely, if~$\\sigma$ is a\ntwisted Baxter permutation, apply {\\sc ExtractBaxter} on~${\\sf P}(\\sigma)$\nto obtain its corresponding Baxter permutation.\n\\medskip\n\nIn the same way, we can compute a bijection between Baxter permutations\nand anti-twisted Baxter permutations using {\\sc ExtractMax} instead of\n{\\sc ExtractMin}. Moreover, these algorithms give a bijection between\ntwisted Baxter permutations and anti-twisted Baxter permutations:\nIf~$\\sigma$ is a twisted (resp. anti-twisted) Baxter permutation, apply\n{\\sc ExtractMax} (resp. {\\sc ExtractMin}) on~${\\sf P}(\\sigma)$ to obtain\nits corresponding anti-twisted (resp. twisted) Baxter permutation.\n\n\\subsection{Definition and correctness of the iterative insertion algorithm}\nIn what follows, we shall revise our ${\\sf P}$-symbol algorithm that we have\npresented in Section~\\ref{subsec:PSymbBaxter} to make it iterative. Indeed,\nwe propose an insertion algorithm such that, for any word~$u$ such that\n${\\sf P}(u) = (T_L, T_R)$ and any letter~${\\tt a}$, the insertion of~${\\tt a}$\ninto~$(T_L, T_R)$ is the pair of twin binary trees~${\\sf P}(u{\\tt a})$. This,\nbesides being in agreement with the usual Robinson-Schensted-like algorithms,\nhas the merit to allow to compute in the Baxter monoid. Indeed, this gives\na simple way to compute the concatenation of two words~$u$ and~$v$ under\nthe Baxter congruence simply by inserting the letters of the word~$uv$\ninto the pair~$(\\perp, \\perp)$. Note that one can compute the\nproduct of two pairs of twin binary trees~$(T_L, T_R)$ and~$(T'_L, T'_R)$\nby computing a word~$u'$ that belongs to the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class\nof~$(T'_L, T'_R)$ by applying the algorithm {\\sc ExtractMin} (or\n{\\sc ExtractBaxter}) with~$(T'_L, T'_R)$ as input, and then, by inserting\nthe letters of~$u'$ from left to right into~$(T_L, T_R)$.\n\n\\subsubsection{Root insertion in binary search trees}\nLet~$T$ be an $A$-labeled right binary search tree and~${\\tt b}$ a letter of~$A$.\nThe \\emph{lower restricted binary tree} of~$T$ compared to~${\\tt b}$,\nnamely~$T_{\\leq {\\tt b}}$, is the right binary search tree uniquely made of\nthe nodes~$x$ of~$T$ labeled by letters~${\\tt a}$ satisfying~${\\tt a} \\leq {\\tt b}$\nand such that for all nodes~$x$ and~$y$ of~$T_{\\leq {\\tt b}}$, if~$x$ is\nancestor of~$y$ in~$T_{\\leq {\\tt b}}$, then~$x$ is also ancestor of~$y$ in~$T$.\nIn the same way, we define the \\emph{higher restricted binary tree}\nof~$T$ compared to~${\\tt b}$, namely~$T_{> {\\tt b}}$ (see Figure~\\ref{fig:ExempleABRestreints}).\n\\begin{figure}[ht]\n    \\centering\n    \\begin{equation*}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqFonce](1)at(1,-1){$1$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqFonce](2)at(2,-3){$2$};\n            \\node[Noeud,EtiqClair](3)at(3,-4){$3$};\n            \\draw[Arete](2)--(3);\n            \\node[Noeud,EtiqClair](4)at(4,-2){$3$};\n            \\draw[Arete](4)--(2);\n            \\draw[Arete](1)--(4);\n            \\node[Noeud,EtiqClair](5)at(5,0){$4$};\n            \\draw[Arete](5)--(1);\n            \\node[Noeud,EtiqClair](6)at(6,-1){$5$};\n            \\draw[Arete](5)--(6);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\qquad\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,-1){$1$};\n            \\node[Noeud,EtiqFonce](1)at(1,0){$1$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqFonce](2)at(2,-1){$2$};\n            \\draw[Arete](1)--(2);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\qquad\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$3$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$3$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,0){$4$};\n            \\draw[Arete](2)--(1);\n            \\node[Noeud,EtiqClair](3)at(3,-1){$5$};\n            \\draw[Arete](2)--(3);\n        \\end{tikzpicture}}\n        \\end{split}\n    \\end{equation*}\n    \\caption{A right binary search tree~$T$, $T_{\\leq 2}$ and~$T_{>2}$.}\n    \\label{fig:ExempleABRestreints}\n\\end{figure}\n\\medskip\n\nLet~$T$ be an $A$-labeled right binary search tree and~${\\tt a}$ a letter\nof~$A$. The \\emph{root insertion} of~${\\tt a}$ into~$T$ consists in modifying~$T$\nso that the root of~$T$ is a new node labeled by~${\\tt a}$, its left subtree\nis~$T_{\\leq {\\tt a}}$ and its right subtree is~$T_{> {\\tt a}}$.\n\n\\subsubsection{The iterative insertion algorithm}\n\n\\begin{Definition} \\label{def:BaxterPQSymboleIt}\n    Let~$(T_L, T_R)$ be an $A$-labeled pair of twin binary search trees\n    and~${\\tt a}$ be a letter. The \\emph{insertion} of~${\\tt a}$ into~$(T_L, T_R)$\n    consists in making a leaf insertion of~${\\tt a}$ into~$T_L$ and a root\n    insertion of~${\\tt a}$ into~$T_R$. The \\emph{iterative Baxter ${\\sf P}$-symbol}\n    (or simply \\emph{iterative ${\\sf P}$-symbol} if the context is clear)\n    of a word~$u \\in A^*$ is the pair ${\\sf P}(u) = (T_L, T_R)$ computed\n    by iteratively inserting the letters of~$u$, from left to right, into\n    $(\\perp, \\perp)$. The \\emph{iterative Baxter ${\\sf Q}$-symbol}\n    (or simply \\emph{iterative ${\\sf Q}$-symbol} if the context is clear)\n    of~$u \\in A^*$ is the pair ${\\sf Q}(u) = (S_L, S_R)$ of same shape\n    as~${\\sf P}(u)$ and such that each node is labeled by its date of\n    creation in~${\\sf P}(u)$.\n\\end{Definition}\n\nFigure~\\ref{fig:ExemplePQSymbole} shows, step by step, the computation\nof the iterative Baxter ${\\sf P}$ and ${\\sf Q}$-symbols of a word.\n\\begin{figure}[ht]\n    \\centering\n    \\begin{equation*}\n    \\begin{split}\\perp \\perp\\end{split}\n    \\begin{split}\\quad \\xrightarrow{\\hspace{.5em}2\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqFonce](0)at(0,0){$2$};\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqFonce](0)at(0,0){$2$};\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split}\\quad \\xrightarrow{\\hspace{.5em}4\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,0){$2$};\n        \\node[Noeud,EtiqFonce](1)at(1.0,-1){$4$};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-1){$2$};\n        \\node[Noeud,EtiqFonce](1)at(1,0){$4$};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split}\\quad \\xrightarrow{\\hspace{.5em}1\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqFonce](0)at(0,-1){$1$};\n        \\node[Noeud,EtiqClair](1)at(1,0){$2$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2,-1){$4$};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqFonce](0)at(0,0){$1$};\n        \\node[Noeud,EtiqClair](1)at(1,-2){$2$};\n        \\node[Noeud,EtiqClair](2)at(2,-1){$4$};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\end{equation*}\n    \\begin{equation*}\n    \\begin{split}\\xrightarrow{\\hspace{.5em}5\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-1){$1$};\n        \\node[Noeud,EtiqClair](1)at(1,0){$2$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2,-1){$4$};\n        \\node[Noeud,EtiqFonce](3)at(3,-2){$5$};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-1){$1$};\n        \\node[Noeud,EtiqClair](1)at(1,-3){$2$};\n        \\node[Noeud,EtiqClair](2)at(2,-2){$4$};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n        \\node[Noeud,EtiqFonce](3)at(3,0){$5$};\n        \\draw[Arete](3)--(0);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split}\\quad \\xrightarrow{\\hspace{.5em}2\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$1$};\n        \\node[Noeud,EtiqClair](1)at(1.0,0){$2$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqFonce](2)at(2.0,-2){$2$};\n        \\node[Noeud,EtiqClair](3)at(3.0,-1){$4$};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud,EtiqClair](4)at(4.0,-2){$5$};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$1$};\n        \\node[Noeud,EtiqClair](1)at(1.0,-2){$2$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqFonce](2)at(2.0,0){$2$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqClair](3)at(3.0,-2){$4$};\n        \\node[Noeud,EtiqClair](4)at(4.0,-1){$5$};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\end{equation*}\n    \\begin{equation*}\n    \\begin{split}\\xrightarrow{\\hspace{.5em}5\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$1$};\n        \\node[Noeud,EtiqClair](1)at(1.0,0){$2$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2.0,-2){$2$};\n        \\node[Noeud,EtiqClair](3)at(3.0,-1){$4$};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud,EtiqClair](4)at(4.0,-2){$5$};\n        \\node[Noeud,EtiqFonce](5)at(5.0,-3){$5$};\n        \\draw[Arete](4)--(5);\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-2){$1$};\n        \\node[Noeud,EtiqClair](1)at(1.0,-3){$2$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqClair](2)at(2.0,-1){$2$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqClair](3)at(3.0,-3){$4$};\n        \\node[Noeud,EtiqClair](4)at(4.0,-2){$5$};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n        \\node[Noeud,EtiqFonce](5)at(5.0,0){$5$};\n        \\draw[Arete](5)--(2);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split}\\xrightarrow{\\hspace{.5em}3\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$1$};\n        \\node[Noeud,EtiqClair](1)at(1.0,0){$2$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2.0,-2){$2$};\n        \\node[Noeud,EtiqFonce](3)at(3.0,-3){$3$};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud,EtiqClair](4)at(4.0,-1){$4$};\n        \\draw[Arete](4)--(2);\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$5$};\n        \\node[Noeud,EtiqClair](6)at(6.0,-3){$5$};\n        \\draw[Arete](5)--(6);\n        \\draw[Arete](4)--(5);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-2){$1$};\n        \\node[Noeud,EtiqClair](1)at(1.0,-3){$2$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqClair](2)at(2.0,-1){$2$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqFonce](3)at(3.0,0){$3$};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud,EtiqClair](4)at(4.0,-3){$4$};\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$5$};\n        \\draw[Arete](5)--(4);\n        \\node[Noeud,EtiqClair](6)at(6.0,-1){$5$};\n        \\draw[Arete](6)--(5);\n        \\draw[Arete](3)--(6);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split} = {\\sf P}(u) \\end{split}\n    \\end{equation*}\n    %\n    \\vspace{2em}\n    %\n    \\begin{equation*}\n    \\begin{split}\\perp \\perp \\end{split}\n    \\begin{split}\\quad \\xrightarrow{\\hspace{.5em}2\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqFonce](0)at(0,0){$1$};\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqFonce](0)at(0,0){$1$};\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split}\\quad \\xrightarrow{\\hspace{.5em}4\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,0){$1$};\n        \\node[Noeud,EtiqFonce](1)at(1.0,-1){$2$};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-1){$1$};\n        \\node[Noeud,EtiqFonce](1)at(1,0){$2$};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split}\\quad \\xrightarrow{\\hspace{.5em}1\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqFonce](0)at(0,-1){$3$};\n        \\node[Noeud,EtiqClair](1)at(1,0){$1$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2,-1){$2$};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqFonce](0)at(0,0){$3$};\n        \\node[Noeud,EtiqClair](1)at(1,-2){$1$};\n        \\node[Noeud,EtiqClair](2)at(2,-1){$2$};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\end{equation*}\n    \\begin{equation*}\n    \\begin{split}\\xrightarrow{\\hspace{.5em}5\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-1){$3$};\n        \\node[Noeud,EtiqClair](1)at(1,0){$1$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2,-1){$2$};\n        \\node[Noeud,EtiqFonce](3)at(3,-2){$4$};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-1){$3$};\n        \\node[Noeud,EtiqClair](1)at(1,-3){$1$};\n        \\node[Noeud,EtiqClair](2)at(2,-2){$2$};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n        \\node[Noeud,EtiqFonce](3)at(3,0){$4$};\n        \\draw[Arete](3)--(0);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split}\\quad \\xrightarrow{\\hspace{.5em}2\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$3$};\n        \\node[Noeud,EtiqClair](1)at(1.0,0){$1$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqFonce](2)at(2.0,-2){$5$};\n        \\node[Noeud,EtiqClair](3)at(3.0,-1){$2$};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud,EtiqClair](4)at(4.0,-2){$4$};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$3$};\n        \\node[Noeud,EtiqClair](1)at(1.0,-2){$1$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqFonce](2)at(2.0,0){$5$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqClair](3)at(3.0,-2){$2$};\n        \\node[Noeud,EtiqClair](4)at(4.0,-1){$4$};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\end{equation*}\n    \\begin{equation*}\n    \\begin{split}\\xrightarrow{\\hspace{.5em}5\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$3$};\n        \\node[Noeud,EtiqClair](1)at(1.0,0){$1$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2.0,-2){$5$};\n        \\node[Noeud,EtiqClair](3)at(3.0,-1){$2$};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud,EtiqClair](4)at(4.0,-2){$4$};\n        \\node[Noeud,EtiqFonce](5)at(5.0,-3){$6$};\n        \\draw[Arete](4)--(5);\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-2){$3$};\n        \\node[Noeud,EtiqClair](1)at(1.0,-3){$1$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqClair](2)at(2.0,-1){$5$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqClair](3)at(3.0,-3){$2$};\n        \\node[Noeud,EtiqClair](4)at(4.0,-2){$4$};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n        \\node[Noeud,EtiqFonce](5)at(5.0,0){$6$};\n        \\draw[Arete](5)--(2);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split}\\xrightarrow{\\hspace{.5em}3\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$3$};\n        \\node[Noeud,EtiqClair](1)at(1.0,0){$1$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2.0,-2){$5$};\n        \\node[Noeud,EtiqFonce](3)at(3.0,-3){$7$};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud,EtiqClair](4)at(4.0,-1){$2$};\n        \\draw[Arete](4)--(2);\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$4$};\n        \\node[Noeud,EtiqClair](6)at(6.0,-3){$6$};\n        \\draw[Arete](5)--(6);\n        \\draw[Arete](4)--(5);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-2){$3$};\n        \\node[Noeud,EtiqClair](1)at(1.0,-3){$1$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqClair](2)at(2.0,-1){$5$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqFonce](3)at(3.0,0){$7$};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud,EtiqClair](4)at(4.0,-3){$2$};\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$4$};\n        \\draw[Arete](5)--(4);\n        \\node[Noeud,EtiqClair](6)at(6.0,-1){$6$};\n        \\draw[Arete](6)--(5);\n        \\draw[Arete](3)--(6);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split} = {\\sf Q}(u) \\end{split}\n    \\end{equation*}\n    \\caption{Steps of the computation of the ${\\sf P}$-symbol and the\n    ${\\sf Q}$-symbol of~$u := 2415253$.}\n    \\label{fig:ExemplePQSymbole}\n\\end{figure}\n\n\\subsubsection{Correctness of the iterative insertion algorithm}\nTo show that the iterative version of the Baxter ${\\sf P}$-symbol computes\nthe same labeled pair of twin binary trees than its non-iterative version,\nwe need the following lemma.\n\\begin{Lemme} \\label{lem:SensInsertion}\n    Let $u \\in A^*$. Let~$T$ be the right binary search tree obtained by\n    root insertions of the letters of~$u$, from left to right. Let~$T'$\n    be the right binary search tree obtained by leaf insertions of the\n    letters of~$u$, from right to left. Then,~$T = T'$.\n\\end{Lemme}\n\\begin{proof}\n    Let us proceed by induction on~$|u|$. If $u = \\epsilon$, the lemma is\n    satisfied. Otherwise, assume that~$u = v {\\tt a}$ where~${\\tt a} \\in A$.\n    Let~$S$ be the right binary search tree obtained by root insertions of the\n    letters of~$v$ from left to right. By induction hypothesis,~$S$ also is\n    the right binary tree obtained by leaf insertions of the letters of~$v$\n    from right to left. The right binary search tree~$T$ obtained by root\n    insertions of~$u$ from left to right satisfies, by definition,\n    $T = S_{\\leq {\\tt a}} \\wedge_{\\tt a} S_{> {\\tt a}}$. The right binary\n    search tree~$T'$ obtained by leaf insertions of~$u$ from right to left\n    satisfies $T' = L' \\wedge_{\\tt a} R'$ where the subtree~$L'$ only depends\n    on the subword $v_{\\leq {\\tt a}} := v_{|]-\\infty, {\\tt a}]}$ and the subtree~$R'$\n    only depends on the subword $v_{> {\\tt a}} := v_{|]{\\tt a}, +\\infty[}$, so that,\n    by induction hypothesis, $L' = S_{\\leq {\\tt a}}$, $R' = S_{> {\\tt a}}$ and\n    thus,~$T = T'$.\n\\end{proof}\n\n\\begin{Proposition} \\label{prop:PSymboleIteratif}\n    For any $u \\in A^*$, the Baxter ${\\sf P}$-symbol of~$u$ and the iterative\n    Baxter ${\\sf P}$-symbol of~$u$ are equal.\n\\end{Proposition}\n\\begin{proof}\n    Let $(T_L, T_R)$ be the ${\\sf P}$-symbol of~$u$ and~$(T'_L, T'_R)$ be\n    the iterative ${\\sf P}$-symbol of~$u$. By definition of these two insertion\n    algorithms, we have~$T_L = T'_L$. Moreover,~$T_R$ is obtained by leaf\n    insertions of the letters of~$u$ from right to left and~$T'_R$ is\n    obtained by root insertions of the letters of~$u$ from left to right.\n    By Lemma~\\ref{lem:SensInsertion}, we have~$T_R = T'_R$.\n\\end{proof}\n\nThe correctness of the iterative version of the ${\\sf Q}$-symbol algorithm\ncomes from the correctness of the iterative ${\\sf P}$-algorithm.\n\n\\section{The Baxter lattice} \\label{sec:TreillisBaxter}\n\n\\subsection{The Baxter lattice congruence}\nRecall that an equivalence relation $\\equiv$ on the elements of a lattice\n$(L, \\wedge, \\vee)$ is a \\emph{lattice congruence} if for all $x, x', y, y' \\in L$,\n$x \\equiv x'$ and~$y \\equiv y'$ imply $x \\wedge y \\equiv x' \\wedge y'$\nand~$x \\vee y \\equiv x' \\vee y'$. The quotient~$L\/_\\equiv$ of~$L$ by~$\\equiv$\nis naturally a lattice. Indeed, by denoting by $\\tau : L \\to L\/_\\equiv$ the\ncanonical projection, the set~$L\/_\\equiv$ is endowed with meet and join\noperations defined by $\\widehat{x} \\wedge \\widehat{y} := \\tau(x \\wedge y)$\nand $\\widehat{x} \\vee \\widehat{y} := \\tau(x \\vee y)$ for all\n$\\widehat{x}, \\widehat{y} \\in L\/_\\equiv$ where~$x$ and~$y$ are any\nelements of~$L$ such that~$\\tau(x) = \\widehat{x}$ and~$\\tau(y) = \\widehat{y}$.\n\\medskip\n\nLattices congruences admit the following very useful order-theoretic\ncharacterization~\\cite{CS98,Rea05}. An equivalence relation~$\\equiv$\non the elements of a lattice~$(L, \\wedge, \\vee)$ seen as a poset~$(L, \\leq)$\nis a lattice congruence is the following three conditions hold.\n\\begin{enumerate}[label = (L\\arabic*)]\n    \\item Every $\\equiv$-equivalence class is an interval of~$L$;\n    \\label{item:LatticeCong1}\n    \\item For any $x, y \\in L$, if $x \\leq y$ then $x {\\!\\downarrow} \\leq y {\\!\\downarrow}$\n    where $x {\\!\\downarrow}$ is the maximal element of the $\\equiv$-equivalence\n    class of~$x$; \\label{item:LatticeCong2}\n    \\item For any $x, y \\in L$, if $x \\leq y$ then $x {\\!\\uparrow} \\leq y {\\!\\uparrow}$\n    where~$x {\\!\\uparrow}$ is the minimal element of the $\\equiv$-equivalence\n    class of~$x$. \\label{item:LatticeCong3}\n\\end{enumerate}\n\nFor any permutation~$\\sigma$, let us denote by~$\\sigma {\\!\\downarrow}$\n(resp.~$\\sigma {\\!\\uparrow}$) the maximal (resp. minimal) permutation of\nthe ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class of~$\\sigma$ for the permutohedron order.\nNote by Proposition~\\ref{prop:EquivBXInter} that~$\\sigma {\\!\\downarrow}$\nand~$\\sigma {\\!\\uparrow}$ are well-defined.\n\n\\begin{Theoreme} \\label{thm:OrdreBaxterMinMax}\n    The Baxter equivalence relation is a lattice congruence of the permutohedron.\n\\end{Theoreme}\n\\begin{proof}\n    By Proposition~\\ref{prop:EquivBXInter}, any Baxter equivalence class\n    of permutations is an interval of the permutohedron, so that\n    \\ref{item:LatticeCong1} checks out. One just has to show that~${\\:\\equiv_{\\operatorname{B}}\\:}$\n    satisfies \\ref{item:LatticeCong2} and \\ref{item:LatticeCong3}.\n\n    Let~$\\sigma$ and~$\\nu$ two permutations such that~$\\sigma {\\:\\leq_{\\operatorname{P}}\\:} \\nu$.\n    Let us show that $\\sigma {\\!\\downarrow} {\\:\\leq_{\\operatorname{P}}\\:} \\nu {\\!\\downarrow}$.\n    It is enough to check the property when~$\\nu = \\sigma s_i$ where~$s_i$\n    is an elementary transposition and~$i$ is not a descent of~$\\sigma$.\n    If~$\\sigma = \\sigma {\\!\\downarrow}$, then\n    $\\sigma {\\!\\downarrow} {\\:\\leq_{\\operatorname{P}}\\:} \\nu {\\:\\leq_{\\operatorname{P}}\\:} \\nu {\\!\\downarrow}$\n    and the property holds. Otherwise, by Lemma~\\ref{lem:DiagHasseEqBX},\n    there exists an elementary transposition~$s_j$ and a permutation~$\\pi$\n    such that~$\\pi$ and~$\\sigma$ are ${\\:\\rightleftarrows_{\\operatorname{B}}\\:}$-adjacent,~$\\pi = \\sigma s_j$\n    and~$\\sigma {\\:\\leq_{\\operatorname{P}}\\:} \\pi$. It then remains to prove that there exists\n    a permutation~$\\mu$ such that~$\\nu {\\:\\equiv_{\\operatorname{B}}\\:} \\mu$ and~$\\pi {\\:\\leq_{\\operatorname{P}}\\:} \\mu$.\n    Indeed, this leads to show, by applying iteratively this reasoning,\n    that~$\\sigma {\\!\\downarrow}$ is smaller than a permutation belonging to the\n    ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class of~$\\nu$ for the permutohedron order and\n    hence, by transitivity, that~$\\sigma {\\!\\downarrow} {\\:\\leq_{\\operatorname{P}}\\:} \\nu {\\!\\downarrow}$.\n    We have four cases:\n    \\begin{enumerate}[label = {\\bf Case \\arabic*:}, fullwidth]\n        \\item If $j \\leq i - 2$, $\\sigma$ is of the form\n        $\\sigma = u \\, {\\tt a} {\\tt b} \\, v \\, {\\tt c} {\\tt d} \\, w$ where~$u$, $v$,\n        and~$w$ are some words and~${\\tt a}$ (resp.~${\\tt c}$) is the $j$-th\n        (resp. $i$-th) letter of~$\\sigma$. One has~${\\tt a} < {\\tt b}$\n        and~${\\tt c} < {\\tt d}$ since~$i$ and~$j$ are not descents of~$\\sigma$.\n        We have $\\nu = u \\, {\\tt a} {\\tt b} \\, v \\, {\\tt d} {\\tt c} \\, w$ and\n        $\\nu s_j = u \\, {\\tt b} {\\tt a} \\, v \\, {\\tt d} {\\tt c} \\, w =: \\mu$. Moreover,\n        since~$\\pi {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\sigma$, there are some letters~${\\tt x} \\in \\operatorname{Alph}(u)$\n        and ${\\tt y} \\in \\operatorname{Alph}(v \\, {\\tt c} {\\tt d} \\, w)$ such that ${\\tt a} < {\\tt x}, {\\tt y} < {\\tt b}$.\n        Thus,~$\\mu {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu$. Finally, since\n        $\\pi = u \\, {\\tt b} {\\tt a} \\, v \\, {\\tt c} {\\tt d} \\, w$, $\\pi {\\:\\leq_{\\operatorname{P}}\\:} \\mu$,\n        so that~$\\mu$ is appropriate.\n        \\item If $j \\geq i + 2$, this is analogous to the previous case.\n        \\item If $j = i + 1$, $\\sigma$ is of the form\n        $\\sigma = u \\, {\\tt a} {\\tt b} {\\tt c} \\, v$ where~$u$ and~$v$ are some words\n        and~${\\tt a}$ is the $i$-th letter of~$\\sigma$. One has ${\\tt a} < {\\tt b} < {\\tt c}$\n        since~$i$ and~$j$ are not descents of~$\\sigma$. Since~$\\sigma {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\pi$,\n        there are some letters~${\\tt x} \\in \\operatorname{Alph}(u)$ and~${\\tt y} \\in \\operatorname{Alph}(v)$\n        such that ${\\tt b} < {\\tt x}, {\\tt y} < {\\tt c}$. Thus, since $\\nu = u \\, {\\tt b} {\\tt a} {\\tt c} \\, v$\n        and ${\\tt a} < {\\tt b} < {\\tt x}, {\\tt y} < {\\tt c}$, we have\n        $\\nu s_j = u \\, {\\tt b} {\\tt c} {\\tt a} \\, v {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu$. Moreover,\n        $\\nu s_j s_i = u \\, {\\tt c} {\\tt b} {\\tt a} \\, v =: \\mu$ and\n        $\\nu s_j {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu s_j s_i$ since ${\\tt b} < {\\tt x}, {\\tt y} < {\\tt c}$ and\n        thus,~$\\mu {\\:\\equiv_{\\operatorname{B}}\\:} \\nu$. Finally, since $\\pi = u \\, {\\tt a} {\\tt c} {\\tt b} \\, v$,\n        we have~$\\pi {\\:\\leq_{\\operatorname{P}}\\:} \\mu$, and hence~$\\mu$ is appropriate.\n        \\item If $j = i - 1$, this is analogous to the previous case.\n    \\end{enumerate}\n    Hence, the Baxter equivalence relation satisfies \\ref{item:LatticeCong2}.\n    The proof that~${\\:\\equiv_{\\operatorname{B}}\\:}$ satisfies \\ref{item:LatticeCong3} is analogous.\n\\end{proof}\n\n\\subsection{A lattice structure over the set of pairs of twin binary trees}\nRecall that by Theorem~\\ref{thm:BijectionEquivBXPermuBX}, the Baxter\nequivalence classes of permutations are in correspondence with unlabeled\npairs of twin binary trees. Thus, the quotient of the permutohedron of\norder~$n$ by the Baxter congruence is a lattice $(\\mathcal{T}\\mathcal{B}\\mathcal{T}_n, {\\:\\leq_{\\operatorname{B}}\\:})$ where\nthe \\emph{Baxter order relation}~${\\:\\leq_{\\operatorname{B}}\\:}$ satisfies, for any~$J_0, J_1 \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}_n$,\n\\begin{equation}\n    J_0 {\\:\\leq_{\\operatorname{B}}\\:} J_1 \\qquad \\mbox{if and only if} \\qquad\n    \\substack{\\mbox{there are $\\sigma, \\nu \\in \\mathfrak{S}_n$ such that} \\\\[.3em]\n              \\mbox{$\\sigma {\\:\\leq_{\\operatorname{P}}\\:} \\nu$, ${\\sf P}\\left(\\sigma\\right) = J_0$\n                    and ${\\sf P}\\left(\\nu\\right) = J_1$}.}\n\\end{equation}\nLet us call \\emph{Baxter lattice} the lattice $(\\mathcal{T}\\mathcal{B}\\mathcal{T}_n, {\\:\\leq_{\\operatorname{B}}\\:})$.\nFigure~\\ref{fig:TreillisBaxter} shows the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes\nin the permutohedron of order $4$ that form the Baxter lattice~$(\\mathcal{T}\\mathcal{B}\\mathcal{T}_4, {\\:\\leq_{\\operatorname{B}}\\:})$.\n\\begin{figure}[ht]\n    \\centering\n    \\begin{tikzpicture}\n        \\draw[Classe,rotate = -45] (1.4,-2.1) ellipse (5mm and 12mm);\n        \\draw[Classe,rotate = -45] (2.8,-2.1) ellipse (5mm and 12mm);\n        \\node[Classe] (1234) at (0, 0) {$1234$};\n        \\node[Classe] (2134) at (-2, -1) {$2134$};\n        \\node[Classe] (1243) at (2, -1) {$1243$};\n        \\node[Classe] (1324) at (0, -1) {$1324$};\n        \\node[Classe] (2314) at (-4, -2) {$2314$};\n        \\node[] (2143) at (0, -2) {$2143$};\n        \\node[Classe] (3124) at (-2, -2) {$3124$};\n        \\node[Classe] (1423) at (4, -2) {$1423$};\n        \\node[Classe] (1342) at (2, -2) {$1342$};\n        \\node[Classe] (2341) at (-5, -3) {$2341$};\n        \\node[Classe] (3214) at (-3, -3) {$3214$};\n        \\node[] (2413) at (-1, -3) {$2413$};\n        \\node[] (3142) at (1, -3) {$3142$};\n        \\node[Classe] (4123) at (3, -3) {$4123$};\n        \\node[Classe] (1432) at (5, -3) {$1432$};\n        \\node[Classe] (3241) at (-4, -4) {$3241$};\n        \\node[Classe] (2431) at (-2, -4) {$2431$};\n        \\node[] (3412) at (0, -4) {$3412$};\n        \\node[Classe] (4213) at (2, -4) {$4213$};\n        \\node[Classe] (4132) at (4, -4) {$4132$};\n        \\node[Classe] (3421) at (-2, -5) {$3421$};\n        \\node[Classe] (4231) at (0, -5) {$4231$};\n        \\node[Classe] (4312) at (2, -5) {$4312$};\n        \\node[Classe] (4321) at (0, -6) {$4321$};\n        \\draw[Arete] (1234) -- (2134);\n        \\draw[Arete] (1234) -- (1243);\n        \\draw[Arete] (1234) -- (1324);\n        \\draw[Arete] (2134) -- (2314);\n        \\draw[Arete] (2134) -- (2143);\n        \\draw[Arete] (1243) -- (2143);\n        \\draw[Arete] (1243) -- (1423);\n        \\draw[Arete] (1324) -- (3124);\n        \\draw[Arete] (1324) -- (1342);\n        \\draw[Arete] (2314) -- (2341);\n        \\draw[Arete] (2314) -- (3214);\n        \\draw[Arete] (2143) -- (2413);\n        \\draw[Arete] (3124) -- (3214);\n        \\draw[Arete] (3124) -- (3142);\n        \\draw[Arete] (1423) -- (4123);\n        \\draw[Arete] (1423) -- (1432);\n        \\draw[Arete] (1342) -- (3142);\n        \\draw[Arete] (1342) -- (1432);\n        \\draw[Arete] (2341) -- (3241);\n        \\draw[Arete] (3214) -- (3241);\n        \\draw[Arete] (2341) -- (2431);\n        \\draw[Arete] (2413) -- (2431);\n        \\draw[Arete] (2413) -- (4213);\n        \\draw[Arete] (3142) -- (3412);\n        \\draw[Arete] (4123) -- (4213);\n        \\draw[Arete] (4123) -- (4132);\n        \\draw[Arete] (1432) -- (4132);\n        \\draw[Arete] (3241) -- (3421);\n        \\draw[Arete] (2431) -- (4231);\n        \\draw[Arete] (3412) -- (3421);\n        \\draw[Arete] (3412) -- (4312);\n        \\draw[Arete] (4213) -- (4231);\n        \\draw[Arete] (4132) -- (4312);\n        \\draw[Arete] (3421) -- (4321);\n        \\draw[Arete] (4231) -- (4321);\n        \\draw[Arete] (4312) -- (4321);\n    \\end{tikzpicture}\n    \\caption{The permutohedron of order~$4$ cut into Baxter equivalence classes.}\n    \\label{fig:TreillisBaxter}\n\\end{figure}\n\n\\subsection{Covering relations of the Baxter lattice}\nLet us describe the covering relations of the lattice $(\\mathcal{T}\\mathcal{B}\\mathcal{T}_n, {\\:\\leq_{\\operatorname{B}}\\:})$\nin terms of operations on pairs of twin binary trees. Consider a Baxter\nequivalence class~$\\widehat{\\sigma}$ of permutations encoded by a pair\nof twin binary trees $(T_L, T_R)$. Let~$\\sigma$ by the maximal element\nof~$\\widehat{\\sigma}$. If~$i$ is a descent of~$\\sigma$, the permutation~$\\sigma s_i$\nis not in~$\\widehat{\\sigma}$, and, by definition of the Baxter lattice,\nthe pair of twin binary trees ${\\sf P}(\\sigma s_i) =: (T'_L, T'_R)$\ncovers~$(T_L, T_R)$. The permutations~$\\sigma$ and~$\\sigma s_i$ satisfy\n\\begin{equation}\n    \\sigma = u \\, {\\tt a} {\\tt d} \\, v \\qquad \\mbox{and} \\qquad\n    \\sigma s_i = u \\, {\\tt d} {\\tt a} \\, v,\n\\end{equation}\nwhere~${\\tt a} < {\\tt d}$. There are three cases whether the factor~$u$ or~$v$\ncontains a letter~${\\tt b}$ satisfying ${\\tt a} < {\\tt b} < {\\tt d}$. Since the quotient\nof the permutohedron by the sylvester congruence is the Tamari\nlattice~\\cite{HNT05} and that covering relations in the Tamari lattice are\nbinary tree rotations, the covering relations of the Baxter lattice are\nthe following:\n\\begin{enumerate}[label = (C\\arabic*)]\n    \\item If there is a letter~${\\tt b}$ in~$v$ such that ${\\tt a} < {\\tt b} < {\\tt d}$,\n    then~$T'_R = T_R$ and~$T'_L$ is obtained from~$T_L$ by performing a\n    left rotation that does not change its canopy;\n    \\item If there is a letter~${\\tt b}$ in~$u$ such that ${\\tt a} < {\\tt b} < {\\tt d}$,\n    then~$T'_L = T_L$ and~$T'_R$ is obtained from~$T_R$ by performing a\n    right rotation that does not change its canopy;\n    \\item If for any letter~${\\tt b}$ of~$u$ and~$v$, one has ${\\tt b} < {\\tt a}$ or\n    ${\\tt d} < {\\tt b}$, then~$T'_L$ (resp.~$T'_R$) is obtained from~$T_L$\n    (resp.~$T_R$) by performing a left (resp. right) rotation that changes\n    its canopy.\n\\end{enumerate}\n\nHence, according to this characterization of the covering relations of\nthe Baxter lattice and the definition of the Tamari lattice, we have,\nfor any pairs of twin binary trees~$(T_L, T_R)$ and~$(T'_L, T'_R)$,\n\\begin{equation} \\label{eq:RelOrdreBX}\n    (T_L, T_R) {\\:\\leq_{\\operatorname{B}}\\:} (T'_L, T'_R) \\qquad \\mbox{if and only if} \\qquad\n    T'_L {\\:\\leq_{\\operatorname{T}}\\:} T_L \\mbox{ and } T_R {\\:\\leq_{\\operatorname{T}}\\:} T'_R .\n\\end{equation}\n\nNote that a right rotation at root~$y$ in a binary tree~$T$ changes its\ncanopy if and only if the right subtree~$B$ of the left child~$x$ of~$y$\nis empty (see Figure~\\ref{fig:Rotation}). Similarly, a left rotation at\nroot~$y$ changes the canopy of~$T$ if and only if the left subtree~$B$\nof~$y$ is empty. Moreover, if~$y$ is the $i$-th node of~$T$, by\nLemma~\\ref{lem:OrientationFeuille}, one can see that~$B$ is the $i$-th\nleaf of~$T$. Hence, the right (resp. left) rotation at root~$y$ changes\nthe orientation of the $i$-th leaf of~$T$ formerly on the right to the\nleft (resp. left to the right).\n\n\\subsection{Twin Tamari diagrams}\nThe purpose of this section is to introduce \\emph{twin Tamari diagrams}.\nThese diagrams are in bijection with pairs of twin binary trees and provide\na useful realization of the Baxter lattice since it appears that testing\nif two twin Tamari diagrams are comparable under the Baxter order relation\nis immediate.\n\n\\subsubsection{Tamari diagrams and the Tamari order relation}\nPallo introduced in~\\cite{Pal86}  words in bijection with binary trees\n(see also~\\cite{Knu06}). We call \\emph{Tamari diagrams} these words and\nto compute the Tamari diagram~$\\operatorname{td}(T)$ of a binary tree~$T$, just label\neach node~$x$ of~$T$ by the number of nodes in the right subtree of~$x$\nand then, consider its inorder reading.\n\\medskip\n\nAny Tamari diagram~$\\delta$ of length~$n$ satisfies the following two\ninequalities:\n\\begin{enumerate}\n    \\item $0 \\leq \\delta_i \\leq n - i$, \\quad for all $1 \\leq i \\leq n$;\n    \\item $\\delta_{i + j} \\leq \\delta_i - j$,\n    \\quad for all $1 \\leq i \\leq n$ and $1 \\leq j \\leq \\delta_i$.\n\\end{enumerate}\n\\medskip\n\nThe main interest of Tamari diagrams is that they offer a very simple way\nto test if two binary trees are comparable in the Tamari lattice~\\cite{Knu06}.\nIndeed, if~$T$ and $T'$ are two binary trees with~$n$ nodes, one has\n\\begin{equation} \\label{eq:ComparaisonDT}\n    T {\\:\\leq_{\\operatorname{T}}\\:} T' \\qquad \\mbox{if and only if} \\qquad\n    \\operatorname{td}(T)_i \\leq \\operatorname{td}(T')_i \\quad \\mbox{for all $1 \\leq i \\leq n$}.\n\\end{equation}\n\n\\subsubsection{Twin Tamari diagrams and the Baxter order relation}\n\n\\begin{Definition} \\label{def:DTD}\n    A \\emph{twin Tamari diagram} of size~$n$ is a\n    pair~$\\left(\\delta^L, \\delta^R\\right)$ such that~$\\delta^L$ and~$\\delta^R$\n    are Tamari diagrams of length~$n$ and for all index $1 \\leq i \\leq n - 1$,\n    exactly one letter among~$\\delta^L_i$ and~$\\delta^R_i$ is zero.\n\\end{Definition}\n\nNote that we can represent any twin Tamari diagram\n$\\delta := \\left(\\delta^L, \\delta^R\\right)$\nin a more compact way by a word~$\\omega(\\delta)$ were\n\\begin{equation}\n    \\omega(\\delta)_i :=\n    \\begin{cases}\n        - \\delta^L_i & \\mbox{if $\\delta^L_i \\ne 0$}, \\\\\n        \\delta^R_i & \\mbox{otherwise},\n    \\end{cases}\n\\end{equation}\nfor all $1 \\leq i \\leq n$ where~$n$ is the size of~$\\delta$. We graphically\nrepresent a twin Tamari diagram~$\\delta$ by drawing for each index~$i$ a\ncolumn of~$|\\omega(\\delta)_i|$ boxes facing up if~$\\omega(\\delta)_i \\geq 0$\nand facing down otherwise. First twin Tamari diagrams are drawn in\nFigure~\\ref{fig:PremiersDTD}.\n\\begin{figure}[ht]\n    \\begin{equation*}\n        \\begin{split}\\epsilon\\end{split}, \\qquad\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(0.5,0);\n            \\draw (0,0) node[above] {};\n        \\end{tikzpicture}}\n        \\end{split}, \\qquad\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(1.5,0);\n            \\node[CaseBas](0, 0) at (0,-0.5) {};\n            \\draw (0,0) node[above] {};\n            \\draw (1,0) node[above] {};\n        \\end{tikzpicture}}\\end{split}, \\enspace\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(1.5,0);\n            \\node[CaseHaut](0, 0) at (0,0.5) {};\n            \\draw (0,1) node[above] {};\n            \\draw (1,0) node[above] {};\n        \\end{tikzpicture}}\n        \\end{split}, \\qquad\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(2.5,0);\n            \\node[CaseBas](0, 0) at (0,-0.5) {};\n            \\node[CaseBas](0, 1) at (0,-1.5) {};\n            \\draw (0,0) node[above] {};\n            \\node[CaseBas](1, 0) at (1,-0.5) {};\n            \\draw (1,0) node[above] {};\n            \\draw (2,0) node[above] {};\n        \\end{tikzpicture}}\n        \\end{split}, \\enspace\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(2.5,0);\n            \\node[CaseBas](0, 0) at (0,-0.5) {};\n            \\node[CaseBas](0, 1) at (0,-1.5) {};\n            \\draw (0,0) node[above] {};\n            \\node[CaseHaut](1, 0) at (1,0.5) {};\n            \\draw (1,1) node[above] {};\n            \\draw (2,0) node[above] {};\n        \\end{tikzpicture}}\n        \\end{split}, \\enspace\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(2.5,0);\n            \\node[CaseBas](0, 0) at (0,-0.5) {};\n            \\draw (0,0) node[above] {};\n            \\node[CaseHaut](1, 0) at (1,0.5) {};\n            \\draw (1,1) node[above] {};\n            \\draw (2,0) node[above] {};\n        \\end{tikzpicture}}\n        \\end{split}, \\enspace\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(2.5,0);\n            \\node[CaseHaut](0, 0) at (0,0.5) {};\n            \\draw (0,1) node[above] {};\n            \\node[CaseBas](1, 0) at (1,-0.5) {};\n            \\draw (1,0) node[above] {};\n            \\draw (2,0) node[above] {};\n        \\end{tikzpicture}}\n        \\end{split}, \\enspace\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(2.5,0);\n            \\node[CaseHaut](0, 0) at (0,0.5) {};\n            \\node[CaseHaut](0, 1) at (0,1.5) {};\n            \\draw (0,2) node[above] {};\n            \\node[CaseBas](1, 0) at (1,-0.5) {};\n            \\draw (1,0) node[above] {};\n            \\draw (2,0) node[above] {};\n        \\end{tikzpicture}}\n        \\end{split}, \\enspace\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(2.5,0);\n            \\node[CaseHaut](0, 0) at (0,0.5) {};\n            \\node[CaseHaut](0, 1) at (0,1.5) {};\n            \\draw (0,2) node[above] {};\n            \\node[CaseHaut](1, 0) at (1,0.5) {};\n            \\draw (1,1) node[above] {};\n            \\draw (2,0) node[above] {};\n        \\end{tikzpicture}}\n        \\end{split}\n    \\end{equation*}\n    \\caption{First twin Tamari diagrams of size~$0$, $1$, $2$, and~$3$.}\n    \\label{fig:PremiersDTD}\n\\end{figure}\n\n\\begin{Proposition} \\label{prop:DTDBijectionABJ}\n    For any~$n \\geq 0$, the set of twin Tamari diagrams of size~$n$ is\n    in bijection with the set of pairs of twin binary trees with~$n$ nodes.\n    Moreover, this bijection is expressed as follows: If $J := (T_L, T_R)$\n    is a pair of twin binary trees, the twin Tamari diagram in bijection\n    with~$J$ is~$\\operatorname{ttd}(J) := \\left(\\operatorname{td}(T_L), \\operatorname{td}(T_R)\\right)$.\n\\end{Proposition}\n\\begin{proof}\n    Let us show that the application~$\\operatorname{ttd}$ is well-defined, that is\n    $\\operatorname{ttd}(J) =: \\left(\\delta^L, \\delta^R\\right)$ is a twin Tamari diagram.\n    Fix an index $1 \\leq i \\leq n - 1$. By contradiction, assume first\n    that $\\delta^L_i = \\delta^R_i = 0$. By definition of~$\\operatorname{td}$, this\n    implies that the $i$-th nodes of~$T_L$ and~$T_R$ have no right child.\n    Hence, by Lemma~\\ref{lem:OrientationFeuille}, the $i\\!+\\!1$-st leaves\n    of~$T_L$ and~$T_R$ are attached to its $i$-th nodes and are right-oriented.\n    Since $i \\leq n - 1$, these leaves are not the rightmost leaves of~$T_L$\n    and~$T_R$, implying that~$T_L$ and~$T_R$ have not complementary canopies,\n    and hence that~$(T_L, T_R)$ is not a pair of twin binary trees.\n    Assume now that~$\\delta^L_i \\ne 0$ and~$\\delta^R_i \\ne 0$. By definition\n    of~$\\operatorname{td}$, this implies that the $i$-th nodes of~$T_L$ and~$T_R$ have\n    a right child. Hence, by Lemma~\\ref{lem:OrientationFeuille}, the\n    $i\\!+\\!1$-st leaves of~$T_L$ and~$T_R$ are attached to its $i\\!+\\!1$-st\n    nodes and are left-oriented. This implies again that~$(T_L, T_R)$ is\n    not a pair of twin binary trees. Thus,~$\\operatorname{ttd}$ computes twin Tamari diagrams.\n\n    Now, since~$\\operatorname{td}$ is a bijection between the set of binary trees\n    with~$n$ nodes and Tamari diagrams of size~$n$~\\cite{Pal86}, for any\n    twin Tamari diagram~$\\delta$, there is a unique pair of binary\n    trees~$J$ such that~$\\operatorname{ttd}(J) = \\delta$. Using very similar arguments\n    as above, one can prove that the canopies of the trees of~$J$ are\n    complementary, and hence, that $J$ is a pair of twin binary trees.\n\\end{proof}\n\nFigure~\\ref{fig:ExempleBijABJDTD} shows an example of a pair of twin binary\ntrees with the corresponding twin Tamari diagram.\n\\begin{figure}[ht]\n    \\begin{equation*}\n        \\begin{split}\n        \\scalebox{.25}{\n        \\begin{tikzpicture}\n            \\node[Noeud](0)at(0.0,-2){};\n            \\node[Noeud](1)at(1.0,-1){};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud](2)at(2.0,-2){};\n            \\draw[Arete](1)--(2);\n            \\node[Noeud](3)at(3.0,0){};\n            \\draw[Arete](3)--(1);\n            \\node[Noeud](4)at(4.0,-2){};\n            \\node[Noeud](5)at(5.0,-3){};\n            \\draw[Arete](4)--(5);\n            \\node[Noeud](6)at(6.0,-1){};\n            \\draw[Arete](6)--(4);\n            \\node[Noeud](7)at(7.0,-2){};\n            \\draw[Arete](6)--(7);\n            \\draw[Arete](3)--(6);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\enspace\n        \\begin{split}\n        \\scalebox{.25}{\n        \\begin{tikzpicture}\n            \\node[Noeud](0)at(0.0,-1){};\n            \\node[Noeud](1)at(1.0,-4){};\n            \\node[Noeud](2)at(2.0,-3){};\n            \\draw[Arete](2)--(1);\n            \\node[Noeud](3)at(3.0,-4){};\n            \\draw[Arete](2)--(3);\n            \\node[Noeud](4)at(4.0,-2){};\n            \\draw[Arete](4)--(2);\n            \\draw[Arete](0)--(4);\n            \\node[Noeud](5)at(5.0,0){};\n            \\draw[Arete](5)--(0);\n            \\node[Noeud](6)at(6.0,-2){};\n            \\node[Noeud](7)at(7.0,-1){};\n            \\draw[Arete](7)--(6);\n            \\draw[Arete](5)--(7);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\n        \\qquad \\xrightarrow{\\enspace \\operatorname{ttd} \\enspace} \\qquad\n        \\end{split}\n        (01041010, 40100200)\n        \\begin{split}\n        \\qquad \\simeq \\qquad\n        \\end{split}\n        \\begin{split}\n        \\scalebox{.25}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(7.5,0);\n            \\node[CaseHaut](0, 0) at (0,0.53) {};\n            \\node[CaseHaut](0, 1) at (0,1.53) {};\n            \\node[CaseHaut](0, 2) at (0,2.53) {};\n            \\node[CaseHaut](0, 3) at (0,3.53) {};\n            \\node[CaseBas](1, 0) at (1,-0.53) {};\n            \\node[CaseHaut](2, 0) at (2,0.53) {};\n            \\node[CaseBas](3, 0) at (3,-0.53) {};\n            \\node[CaseBas](3, 1) at (3,-1.53) {};\n            \\node[CaseBas](3, 2) at (3,-2.53) {};\n            \\node[CaseBas](3, 3) at (3,-3.53) {};\n            \\node[CaseBas](4, 0) at (4,-0.53) {};\n            \\node[CaseHaut](5, 0) at (5,0.53) {};\n            \\node[CaseHaut](5, 1) at (5,1.53) {};\n            \\node[CaseBas](6, 0) at (6,-0.53) {};\n        \\end{tikzpicture}}\n        \\end{split}\n    \\end{equation*}\n    \\caption{A pair of twin binary trees, the corresponding twin Tamari\n    diagram via the bijection~$\\operatorname{ttd}$ and its graphical representation.}\n    \\label{fig:ExempleBijABJDTD}\n\\end{figure}\n\n\\begin{Proposition} \\label{prop:OrdreBaxterDTD}\n    Let~$J_0$ and~$J_1$ two pairs of twin binary trees with~$n$ nodes.\n    We have\n    \\begin{equation}\n        J_0 {\\:\\leq_{\\operatorname{B}}\\:} J_1\n        \\qquad \\mbox{if and only if} \\qquad\n        \\omega\\left(\\operatorname{ttd}\\left(J_0\\right)\\right)_i \\leq\n        \\omega\\left(\\operatorname{ttd}\\left(J_1\\right)\\right)_i \\quad\n        \\mbox{for all $1 \\leq i \\leq n$}.\n    \\end{equation}\n\\end{Proposition}\n\\begin{proof}\n    This result is a direct consequence of the characterization of the\n    Baxter order relation~(\\ref{eq:RelOrdreBX}) using the Tamari order\n    relation, the characterization furnished by~(\\ref{eq:ComparaisonDT})\n    to compare two binary trees in the Tamari lattice with Tamari diagrams,\n    and the bijection between pairs of twin binary trees and Twin Tamari\n    diagrams provided by Proposition~\\ref{prop:DTDBijectionABJ}.\n\\end{proof}\n\nFigure~\\ref{fig:IntervalleABJ} shows an interval of the Baxter lattice.\n\\begin{figure}[ht]\n    \\centering\n    \\scalebox{.16}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,2){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,0){};\n        \\node[Noeud](4)at(4,1){};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n        \\node[Noeud](5)at(5,0){};\n        \\node[Noeud](6)at(6,1){};\n        \\draw[Arete](6)--(5);\n        \\node[Noeud](7)at(7,0){};\n        \\draw[Arete](6)--(7);\n        \\node[Noeud](8)at(8,2){};\n        \\draw[Arete](8)--(6);\n        \\node[Noeud](9)at(9,1){};\n        \\draw[Arete](8)--(9);\n        \\node[fit=(0) (1) (2) (3) (4) (5) (6) (7) (8) (9)] (J1) {};\n        \\node[Noeud](10)at(-10,-6){};\n        \\node[Noeud](11)at(-9,-7){};\n        \\draw[Arete](10)--(11);\n        \\node[Noeud](12)at(-8,-5){};\n        \\draw[Arete](12)--(10);\n        \\node[Noeud](13)at(-7,-7){};\n        \\node[Noeud](14)at(-6,-6){};\n        \\draw[Arete](14)--(13);\n        \\draw[Arete](12)--(14);\n        \\node[Noeud](15)at(-4,-6){};\n        \\node[Noeud](16)at(-3,-5){};\n        \\draw[Arete](16)--(15);\n        \\node[Noeud](17)at(-2,-7){};\n        \\node[Noeud](18)at(-1,-6){};\n        \\draw[Arete](18)--(17);\n        \\node[Noeud](19)at(0,-7){};\n        \\draw[Arete](18)--(19);\n        \\draw[Arete](16)--(18);\n        \\node[fit=(10) (11) (12) (13) (14) (15) (16) (17) (18) (19)] (J2) {};\n        \\node[Noeud](20)at(11,-7){};\n        \\node[Noeud](21)at(12,-8){};\n        \\draw[Arete](20)--(21);\n        \\node[Noeud](22)at(13,-6){};\n        \\draw[Arete](22)--(20);\n        \\node[Noeud](23)at(14,-7){};\n        \\draw[Arete](22)--(23);\n        \\node[Noeud](24)at(15,-5){};\n        \\draw[Arete](24)--(22);\n        \\node[Noeud](25)at(16,-7){};\n        \\node[Noeud](26)at(17,-6){};\n        \\draw[Arete](26)--(25);\n        \\node[Noeud](27)at(18,-7){};\n        \\draw[Arete](26)--(27);\n        \\node[Noeud](28)at(19,-5){};\n        \\draw[Arete](28)--(26);\n        \\node[Noeud](29)at(20,-6){};\n        \\draw[Arete](28)--(29);\n        \\node[fit=(20) (21) (22) (23) (24) (25) (26) (27) (28) (29)] (J3) {};\n        \\node[Noeud](30)at(0,-15){};\n        \\node[Noeud](31)at(1,-16){};\n        \\draw[Arete](30)--(31);\n        \\node[Noeud](32)at(2,-14){};\n        \\draw[Arete](32)--(30);\n        \\node[Noeud](33)at(3,-15){};\n        \\draw[Arete](32)--(33);\n        \\node[Noeud](34)at(4,-13){};\n        \\draw[Arete](34)--(32);\n        \\node[Noeud](35)at(5,-14){};\n        \\node[Noeud](36)at(6,-13){};\n        \\draw[Arete](36)--(35);\n        \\node[Noeud](37)at(7,-15){};\n        \\node[Noeud](38)at(8,-14){};\n        \\draw[Arete](38)--(37);\n        \\node[Noeud](39)at(9,-15){};\n        \\draw[Arete](38)--(39);\n        \\draw[Arete](36)--(38);\n        \\node[fit=(30) (31) (32) (33) (34) (35) (36) (37) (38) (39)] (J4) {};\n        \\node[Noeud](40)at(0,-25){};\n        \\node[Noeud](41)at(1,-26){};\n        \\draw[Arete](40)--(41);\n        \\node[Noeud](42)at(2,-24){};\n        \\draw[Arete](42)--(40);\n        \\node[Noeud](43)at(3,-23){};\n        \\draw[Arete](43)--(42);\n        \\node[Noeud](44)at(4,-22){};\n        \\draw[Arete](44)--(43);\n        \\node[Noeud](45)at(5,-23){};\n        \\node[Noeud](46)at(6,-22){};\n        \\draw[Arete](46)--(45);\n        \\node[Noeud](47)at(7,-23){};\n        \\node[Noeud](48)at(8,-24){};\n        \\node[Noeud](49)at(9,-25){};\n        \\draw[Arete](48)--(49);\n        \\draw[Arete](47)--(48);\n        \\draw[Arete](46)--(47);\n        \\node[fit=(40) (41) (42) (43) (44) (45) (46) (47) (48) (49)] (J5) {};\n        \\draw [Arete,line width=6pt] (J1)--(J2);\n        \\draw [Arete,line width=6pt] (J1)--(J3);\n        \\draw [Arete,line width=6pt] (J2)--(J4);\n        \\draw [Arete,line width=6pt] (J3)--(J4);\n        \\draw [Arete,line width=6pt] (J4)--(J5);\n    \\end{tikzpicture}}\n    \\qquad \\qquad\n    \\scalebox{.16}{\n    \\begin{tikzpicture}\n       \n        \\draw[Ligne](-0.5,0)--(4.5,0);\n        \\node[CaseBas](c1) at (0,-0.5) {};\n        \\node[CaseHaut](c2) at (1,0.5) {};\n        \\node[CaseBas](c3) at (2,-0.5) {};\n        \\node[CaseBas](c4) at (2,-1.5) {};\n        \\node[CaseHaut](c5) at (3,0.5) {};\n        \\node[fit=(c1) (c2) (c3) (c4) (c5)] (d1) {};\n       \n        \\draw[Ligne](-8.5,-8)--(-3.5,-8);\n        \\node[CaseBas](c6) at (-8,-8.5) {};\n        \\node[CaseHaut](c7) at (-7,-7.5) {};\n        \\node[CaseHaut](c8) at (-7,-6.5) {};\n        \\node[CaseHaut](c9) at (-7,-5.5) {};\n        \\node[CaseBas](c10) at (-6,-8.5) {};\n        \\node[CaseBas](c11) at (-6,-9.5) {};\n        \\node[CaseHaut](c12) at (-5,-7.5) {};\n        \\node[fit=(c6) (c7) (c8) (c9) (c10) (c11) (c12)] (d2) {};\n       \n        \\draw[Ligne](7.5,-8)--(12.5,-8);\n        \\node[CaseBas](c13) at (8,-8.5) {};\n        \\node[CaseHaut](c14) at (9,-7.5) {};\n        \\node[CaseBas](c15) at (10,-8.5) {};\n        \\node[CaseHaut](c16) at (11,-7.5) {};\n        \\node[fit=(c13) (c14) (c15) (c16)] (d3) {};\n       \n        \\draw[Ligne](-0.5,-16)--(4.5,-16);\n        \\node[CaseBas](c17) at (0,-16.5) {};\n        \\node[CaseHaut](c18) at (1,-15.5) {};\n        \\node[CaseHaut](c19) at (1,-14.5) {};\n        \\node[CaseHaut](c20) at (1,-13.5) {};\n        \\node[CaseBas](c21) at (2,-16.5) {};\n        \\node[CaseHaut](c22) at (3,-15.5) {};\n        \\node[fit=(c17) (c18) (c19) (c20) (c21) (c22)] (d4) {};\n       \n        \\draw[Ligne](-0.5,-24)--(4.5,-24);\n        \\node[CaseBas](c23) at (0,-24.5) {};\n        \\node[CaseHaut](c24) at (1,-23.5) {};\n        \\node[CaseHaut](c25) at (1,-22.5) {};\n        \\node[CaseHaut](c26) at (1,-21.5) {};\n        \\node[CaseHaut](c27) at (2,-23.5) {};\n        \\node[CaseHaut](c28) at (2,-22.5) {};\n        \\node[CaseHaut](c29) at (3,-23.5) {};\n        \\node[fit=(c23) (c24) (c25) (c26) (c27) (c28) (c29)] (d5) {};\n       \n        \\draw [Arete,line width=6pt] (d1)--(d2);\n        \\draw [Arete,line width=6pt] (d1)--(d3);\n        \\draw [Arete,line width=6pt] (d2)--(d4);\n        \\draw [Arete,line width=6pt] (d3)--(d4);\n        \\draw [Arete,line width=6pt] (d4)--(d5);\n    \\end{tikzpicture}}\n    \\caption{An interval of the Baxter lattice of order~$5$ where vertices\n    are seen as pairs of twin binary trees and as Twin Tamari diagrams.}\n    \\label{fig:IntervalleABJ}\n\\end{figure}\n\n\\section{The Hopf algebra of pairs of twin binary trees} \\label{sec:AlgebreHopfBaxter}\n\nIn the sequel, all the algebraic structures have a field of characteristic\nzero $\\mathbb{K}$ as ground field.\n\n\\subsection{\\texorpdfstring{The Hopf algebra ${\\bf FQSym}$ and construction\n                            of Hopf subalgebras}\n           {The Hopf algebra FQSym and construction of Hopf subalgebras}}\n\n\\subsubsection{\\texorpdfstring{The Hopf algebra ${\\bf FQSym}$}\n                              {The Hopf algebra FQSym}}\nRecall that the family $\\left\\{{\\bf F}_\\sigma\\right\\}_{\\sigma \\in \\mathfrak{S}}$\nforms the \\emph{fundamental} basis of~${\\bf FQSym}$, the Hopf algebra of Free\nquasi-symmetric functions~\\cite{MR95, DHT02}. Its product and its coproduct\nare defined by\n\\begin{equation}\n    {\\bf F}_\\sigma \\cdot {\\bf F}_\\nu :=\n    \\sum_{\\pi \\; \\in \\; \\sigma \\; \\cshuffle \\; \\nu} {\\bf F}_\\pi,\n\\end{equation}\n\\begin{equation}\n    \\Delta \\left({\\bf F}_\\sigma\\right) :=\n    \\sum_{\\sigma = uv} {\\bf F}_{\\operatorname{std}(u)} \\otimes {\\bf F}_{\\operatorname{std}(v)}.\n\\end{equation}\nFor example,\n\\begin{equation}\\begin{split}\n    {\\bf F}_{132} \\cdot {\\bf F}_{12} & =\n        {\\bf F}_{13245} + {\\bf F}_{13425} + {\\bf F}_{13452} + {\\bf F}_{14325} + {\\bf F}_{14352} \\\\\n    & + {\\bf F}_{14532} + {\\bf F}_{41325} + {\\bf F}_{41352} + {\\bf F}_{41532} + {\\bf F}_{45132},\n\\end{split}\\end{equation}\n\n\\begin{equation}\\begin{split}\n    \\Delta\\left({\\bf F}_{35142}\\right) & =\n        1 \\otimes {\\bf F}_{35142} + {\\bf F}_{1} \\otimes {\\bf F}_{4132} + {\\bf F}_{12} \\otimes {\\bf F}_{132} \\\\\n    & + {\\bf F}_{231} \\otimes {\\bf F}_{21} + {\\bf F}_{2413} \\otimes {\\bf F}_{1} + {\\bf F}_{35142} \\otimes 1.\n\\end{split}\\end{equation}\n\\medskip\n\nSet ${\\bf G}_\\sigma := {\\bf F}_{\\sigma^{-1}}$. Recall that~${\\bf FQSym}$ is isomorphic to\nits dual~${\\bf FQSym}^\\star$ through the map $\\psi : {\\bf FQSym} \\to {\\bf FQSym}^\\star$\ndefined by $\\psi\\left({\\bf F}_\\sigma\\right) := {\\bf F}_{\\sigma^{-1}}^\\star = {\\bf G}_\\sigma^\\star$.\n\\medskip\n\nRecall also that~${\\bf FQSym}$ admits a \\emph{polynomial realization}~\\cite{DHT02},\nthat is an injective algebra morphism\n$r_A : {\\bf FQSym} \\hookrightarrow \\mathbb{K} \\langle A \\rangle$. Furthermore, this map\nshould be compatible with the coalgebra structure in the sense that the\ncoproduct of an element can be computed by taking its image by~$r_A$, and\nthen by applying the \\emph{alphabet doubling trick}~\\cite{DHT02,Hiv07}.\nThis map is defined by\n\\begin{equation} \\label{eq:RealisationFQSym}\n    r_A\\left({\\bf G}_\\sigma\\right) :=\n    \\sum_{\\substack{u \\; \\in \\; A^* \\\\ \\operatorname{std}(u) = \\sigma}} u.\n\\end{equation}\nFor example,\n\\begin{align}\n    r_A\\left({\\bf G}_\\epsilon\\right) & = 1, \\\\\n    r_A\\left({\\bf G}_1\\right)        & = \\sum_i a_i = a_1 + a_2 + a_3 + \\cdots, \\\\\n    r_A\\left({\\bf G}_{231}\\right)    & = \\sum_{k < i \\leq j} a_i a_j a_k\n        = a_2 a_2 a_1 + a_2 a_3 a_1 + a_2 a_4 a_1 + \\cdots.\n\\end{align}\n\n\\subsubsection{\\texorpdfstring{Construction of Hopf subalgebras of ${\\bf FQSym}$}\n                              {Construction of Hopf subalgebras of FQSym}}\nIf~$\\equiv$ is an equivalence relation on~$\\mathfrak{S}$ and~$\\sigma \\in \\mathfrak{S}$,\nlet us denote by~$\\widehat{\\sigma}$ the $\\equiv$-equivalence class of~$\\sigma$.\n\\medskip\n\nThe following theorem contained in an unpublished note of Hivert and\nNzeutchap~\\cite{HN07} (see also~\\cite{DHT02,Hiv07}) shows that an equivalence\nrelation on~$A^*$ satisfying some properties can be used to define Hopf\nsubalgebras of~${\\bf FQSym}$:\n\\begin{Theoreme} \\label{thm:HivertJanvier}\n    Let~$\\equiv$ be an equivalence relation defined on~$A^*$. If~$\\equiv$\n    is a congruence, compatible with the restriction of alphabet intervals\n    and compatible with the destandardization process, then the family\n    $\\left\\{{\\bf P}_{\\widehat{\\sigma}}\\right\\}_{\\widehat{\\sigma} \\in \\mathfrak{S}\/_\\equiv}$\n    defined by\n    \\begin{equation} \\label{eq:EquivFQSym}\n        {\\bf P}_{\\widehat{\\sigma}} :=\n        \\sum_{\\nu \\; \\in \\; \\widehat{\\sigma}} {\\bf F}_\\nu,\n    \\end{equation}\n    spans a Hopf subalgebra of~${\\bf FQSym}$.\n\\end{Theoreme}\nThe compatibility with the destandardization process and with the restriction\nof alphabet intervals imply that for any~${\\bf F}_\\pi$ appearing in a product\n${\\bf P}_{\\widehat{\\sigma}} \\cdot {\\bf P}_{\\widehat{\\nu}}$ and any\npermutation~$\\pi' \\equiv \\pi$, ${\\bf F}_{\\pi'}$ also appears in the product.\nMoreover, the compatibility with the destandardization process and the\nfact that~$\\equiv$ is a congruence imply that for any ${\\bf F}_\\sigma \\otimes {\\bf F}_\\nu$\nappearing in a coproduct~$\\Delta\\left({\\bf P}_{\\widehat{\\pi}}\\right)$ and\nany permutations~$\\sigma' \\equiv \\sigma$ and~$\\nu' \\equiv \\nu$,\n${\\bf F}_{\\sigma'} \\otimes {\\bf F}_{\\nu'}$ also appears in the coproduct.\n\\medskip\n\nIn the sequel, we shall call\n$\\left\\{{\\bf P}_{\\widehat{\\sigma}}\\right\\}_{\\widehat{\\sigma} \\in \\mathfrak{S}\/_\\equiv}$\nthe \\emph{fundamental basis} of the corresponding Hopf subalgebra of~${\\bf FQSym}$.\n\n\\subsection{\\texorpdfstring{Construction of the Hopf algebra ${\\bf Baxter}$}\n                           {Construction of the Hopf algebra Baxter}}\nBy Theorem~\\ref{thm:BijectionEquivBXPermuBX}, the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\nclasses of permutations can be encoded by unlabeled pairs of twin binary\ntrees. Moreover, in the sequel, the ${\\sf P}$-symbols of permutations are\nregarded as unlabeled pairs of twin binary trees since there is only one\nway to label a pair of twin binary trees with a permutation so that it is\na pair of twin binary search trees. Hence, in our graphical representations\nwe will only represent their shape.\n\\medskip\n\nSince by definition~${\\:\\equiv_{\\operatorname{B}}\\:}$ is a congruence, since by\nPropositions~\\ref{prop:CompDestd} and~\\ref{prop:CompRestrSegmAlph},~${\\:\\equiv_{\\operatorname{B}}\\:}$\nsatisfies the conditions of Theorem~\\ref{thm:HivertJanvier}, and since by\nTheorem~\\ref{thm:PSymboleClasses}, the permutations~$\\sigma$ such\nthat~${\\sf P}(\\sigma) = J$ coincide with the Baxter equivalence class\nrepresented by the pair of twin binary trees~$J$, we have the following\ntheorem.\n\\begin{Theoreme} \\label{thm:AlgebreHopfBaxter}\n    The family $\\left\\{{\\bf P}_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$ defined by\n    \\begin{equation}\n        {\\bf P}_J :=\n        \\sum_{\\substack{\\sigma \\; \\in \\; \\mathfrak{S} \\\\ {\\sf P}(\\sigma) = J}}\n        {\\bf F}_\\sigma,\n    \\end{equation}\n    spans a Hopf subalgebra of~${\\bf FQSym}$, namely the Hopf algebra~${\\bf Baxter}$.\n\\end{Theoreme}\n\nFor example,\n\\begin{align}\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}}\n    & =\n    {\\bf F}_{12}, \\\\[1em]\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    & =\n    {\\bf F}_{2143} + {\\bf F}_{2413}, \\\\[1em]\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(1);\n        \\node[Noeud](4)at(4,0){};\n        \\draw[Arete](4)--(3);\n        \\node[Noeud](5)at(5,-1){};\n        \\draw[Arete](4)--(5);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-2){};\n        \\node[Noeud](4)at(4,-3){};\n        \\draw[Arete](3)--(4);\n        \\node[Noeud](5)at(5,-1){};\n        \\draw[Arete](5)--(3);\n        \\draw[Arete](2)--(5);\n    \\end{tikzpicture}}}\n    & =\n    {\\bf F}_{542163} + {\\bf F}_{542613} + {\\bf F}_{546213}.\n\\end{align}\n\\medskip\n\nThe Hilbert series of~${\\bf Baxter}$ is\n\\begin{equation}\n    B(z) := 1 + z + 2z^2 + 6z^3 + 22z^4 + 92z^5 + 422z^6 +\n            2074z^7 + 10754z^8 + 58202z^9 + \\cdots,\n\\end{equation}\nthe generating series of Baxter permutations (sequence~\\Sloane{A001181}\nof~\\cite{SLOANE}).\n\\medskip\n\nBy Theorem~\\ref{thm:HivertJanvier}, the product of~${\\bf Baxter}$ is well-defined.\nWe deduce it from the product of~${\\bf FQSym}$, and, since by Theorem~\\ref{thm:EquivBXBaxter}\nthere is exactly one Baxter permutation in any ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\nclass of permutations, we obtain\n\\begin{equation} \\label{eq:ProduitBaxterP}\n    {\\bf P}_{J_0} \\cdot {\\bf P}_{J_1} =\n        \\sum_{\\substack{{\\sf P}(\\sigma) = J_0, \\; {\\sf P}(\\nu) = J_1 \\\\\n              \\pi \\; \\in \\; \\sigma \\; \\cshuffle \\; \\nu \\; \\cap \\; \\mathfrak{S}^{\\operatorname{B}}}}\n        {\\bf P}_{{\\sf P}(\\pi)}.\n\\end{equation}\nFor example,\n\\begin{equation}\\begin{split}\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}\n    \\cdot\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}}\n    & =\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(1);\n        \\node[Noeud](4)at(4,0){};\n        \\draw[Arete](4)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud](4)at(4,0){};\n        \\draw[Arete](4)--(1);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}} \\\\\n    & +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](4)at(4,0){};\n        \\draw[Arete](4)--(1);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](4)--(2);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\node[Noeud](3)at(3,-3){};\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}.\n\\end{split}\\end{equation}\n\\medskip\n\nIn the same way, we deduce the coproduct of~${\\bf Baxter}$ from the coproduct\nof~${\\bf FQSym}$ and by Theorem~\\ref{thm:EquivBXBaxter}, we obtain\n\\begin{equation}\n    \\Delta \\left({\\bf P}_J\\right) =\n        \\sum_{\\substack{uv \\; \\in \\; \\mathfrak{S} \\\\\n                {\\sf P}(uv) = J \\\\\n                \\sigma := \\operatorname{std}(u), \\; \\nu := \\operatorname{std}(v) \\; \\in \\; \\mathfrak{S}^{\\operatorname{B}}}}\n        {\\bf P}_{{\\sf P}(\\sigma)} \\otimes {\\bf P}_{{\\sf P}(\\nu)}.\n\\end{equation}\nFor example,\n\\begin{equation}\\begin{split}\n    \\Delta \\left(\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\\right)\n    & =\n    1\n    \\otimes\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}} \\\\\n    & +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    \\otimes\n    1.\n\\end{split}\\end{equation}\n\n\\subsection{\\texorpdfstring{Properties of the Hopf algebra ${\\bf Baxter}$}\n                           {Properties of the Hopf algebra Baxter}}\n\n\\subsubsection{A polynomial realization}\nWe deduce a polynomial realization of~${\\bf Baxter}$ from the one of~${\\bf FQSym}$.\nIn this section, we shall use the notation~$J_0 \\simeq J_1$ to say that\nthe labeled pairs of twin binary trees~$J_0$ and~$J_1$ have same shape.\n\\begin{Theoreme}\n    The map $r_A : {\\bf Baxter} \\rightarrow \\mathbb{K} \\langle A \\rangle$ defined by\n    \\begin{equation} \\label{eq:RealisationBaxter}\n        r_A\\left({\\bf P}_J\\right) :=\n            \\sum_{\\substack{u \\; \\in \\; A^* \\\\\n            \\left(\\operatorname{incr}(u), \\; \\operatorname{decr}(u)\\right) \\simeq J}}\n            u,\n    \\end{equation}\n    for any~$J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}$ provides a polynomial realization of~${\\bf Baxter}$.\n\\end{Theoreme}\n\\begin{proof}\n    Let us apply the polynomial realization~$r_A$ of~${\\bf FQSym}$ defined\n    in~(\\ref{eq:RealisationFQSym}) on elements of the fundamental basis\n    of~${\\bf Baxter}$:\n    \\begin{align}\n        r_A\\left({\\bf P}_J\\right) & =\n            \\sum_{\\substack{\\sigma \\; \\in \\; \\mathfrak{S} \\\\\n                {\\sf P}(\\sigma) = J}}\n                r_A({\\bf F}_\\sigma), \\\\\n            & = \\sum_{\\substack{\\sigma \\; \\in \\; \\mathfrak{S} \\\\\n                {\\sf P}(\\sigma^{-1}) = J}}\n                r_A({\\bf G}_\\sigma), \\label{eq:PreuvePR1} \\\\\n            & = \\sum_{\\substack{\\sigma \\; \\in \\; \\mathfrak{S} \\\\\n                \\left(\\operatorname{incr}(\\sigma), \\; \\operatorname{decr}(\\sigma)\\right) \\simeq J}}\n                r_A({\\bf G}_\\sigma), \\label{eq:PreuvePR2} \\\\\n            & = \\sum_{\\substack{\\sigma \\; \\in \\; \\mathfrak{S} \\\\\n                \\left(\\operatorname{incr}(\\sigma), \\; \\operatorname{decr}(\\sigma)\\right) \\simeq J}}\n                \\quad \\sum_{\\substack{u \\; \\in \\; A^* \\\\\n                \\operatorname{std}(u) = \\sigma}} u, \\label{eq:PreuvePR3}\n    \\end{align}\n    The equality between~(\\ref{eq:PreuvePR1}) and~(\\ref{eq:PreuvePR2})\n    follows from Lemma~\\ref{lem:FormeInsIncr}. The equality between~(\\ref{eq:PreuvePR3})\n    and the right member of~(\\ref{eq:RealisationBaxter}) follows from the\n    fact that $\\operatorname{incr}(\\sigma) \\simeq \\operatorname{incr}(u)$ (resp. $\\operatorname{decr}(\\sigma) \\simeq \\operatorname{decr}(u)$)\n    whenever~$\\operatorname{std}(u) = \\sigma$.\n\\end{proof}\n\n\\subsubsection{The dual Hopf algebra}\nWe denote by~$\\left\\{{\\bf P}^\\star_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$ the dual basis\nof the basis $\\left\\{{\\bf P}_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$. The Hopf\nalgebra~${\\bf Baxter}^\\star$, dual of~${\\bf Baxter}$, is a quotient Hopf algebra\nof~${\\bf FQSym}^\\star$. More precisely,\n\\begin{equation}\n    {\\bf Baxter}^\\star = {\\bf FQSym}^\\star \/ I,\n\\end{equation}\nwhere~$I$ is the Hopf ideal of~${\\bf FQSym}^\\star$ spanned by the elements\n$({\\bf F}^\\star_\\sigma - {\\bf F}^\\star_\\nu)$ whenever~$\\sigma {\\:\\equiv_{\\operatorname{B}}\\:} \\nu$.\n\\medskip\n\nLet $\\phi : {\\bf FQSym}^\\star \\twoheadrightarrow {\\bf Baxter}^\\star$ be the canonical\nprojection, mapping~${\\bf F}^\\star_\\sigma$ on~${\\bf P}^\\star_{{\\sf P}(\\sigma)}$.\nBy definition, the product of~${\\bf Baxter}^\\star$ is\n\\begin{equation}\n    {\\bf P}^\\star_{J_0} \\cdot {\\bf P}^\\star_{J_1} =\n    \\phi \\left({\\bf F}^\\star_{\\sigma} \\cdot {\\bf F}^\\star_{\\nu} \\right),\n\\end{equation}\nwhere~$\\sigma$ and~$\\nu$ are any permutations such that ${\\sf P}(\\sigma) = J_0$\nand ${\\sf P}(\\nu) = J_1$. Note that due to the fact that~${\\bf Baxter}^\\star$\nis a quotient of~${\\bf FQSym}^\\star$, the number of terms occurring in a product\n${\\bf P}^\\star_{J_0} \\cdot {\\bf P}^\\star_{J_1}$ only depends on the number~$m$\n(resp.~$n$) of nodes of~$J_0$ (resp.~$J_1$) and is~$\\binom{m + n}{m}$.\nFor example,\n\\begin{equation}\\begin{split}\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-1){};\n        \\node[Noeud,Marque1](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,Marque1](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}\n    \\cdot\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque2](0)at(0,-1){};\n        \\node[Noeud,Marque2](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}}\n    & =\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-1){};\n        \\node[Noeud,Marque1](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,Marque1](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud,Marque2](3)at(3,-1){};\n        \\draw[Arete](3)--(1);\n        \\node[Noeud,Marque2](4)at(4,0){};\n        \\draw[Arete](4)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-1){};\n        \\node[Noeud,Marque1](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,Marque1](3)at(3,0){};\n        \\draw[Arete](3)--(0);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque2](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud,Marque2](4)at(4,0){};\n        \\draw[Arete](4)--(2);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud,Marque1](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n        \\node[Noeud,Marque1](3)at(3,0){};\n        \\draw[Arete](3)--(0);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud,Marque2](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n        \\node[Noeud,Marque2](4)at(4,0){};\n        \\draw[Arete](4)--(1);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque1](2)at(2,-2){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud,Marque1](3)at(3,0){};\n        \\draw[Arete](3)--(1);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque2](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n        \\node[Noeud,Marque2](4)at(4,0){};\n        \\draw[Arete](4)--(0);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-1){};\n        \\node[Noeud,Marque1](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-4){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,Marque1](4)at(4,0){};\n        \\draw[Arete](4)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque2](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud,Marque2](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}} \\\\\n    & +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud,Marque1](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n        \\node[Noeud,Marque1](4)at(4,0){};\n        \\draw[Arete](4)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud,Marque2](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud,Marque2](3)at(3,0){};\n        \\draw[Arete](3)--(1);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud](2)at(2,-4){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\draw[Arete](3)--(1);\n        \\draw[Arete](0)--(3);\n        \\node[Noeud,Marque1](4)at(4,0){};\n        \\draw[Arete](4)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud,Marque2](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque2](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-3){};\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud,Marque1](4)at(4,0){};\n        \\draw[Arete](4)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque2](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,Marque2](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque1](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n        \\node[Noeud,Marque1](4)at(4,0){};\n        \\draw[Arete](4)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque2](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n        \\node[Noeud,Marque2](3)at(3,0){};\n        \\draw[Arete](3)--(0);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,Marque1](2)at(2,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud,Marque1](4)at(4,0){};\n        \\draw[Arete](4)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque2](0)at(0,-1){};\n        \\node[Noeud,Marque2](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}}.\n\\end{split}\\end{equation}\n\\medskip\n\nIn the same way, the coproduct of~${\\bf Baxter}^\\star$ is\n\\begin{equation}\n    \\Delta({\\bf P}_J) =\n    (\\phi \\otimes \\phi)\\left(\\Delta \\left({\\bf F}^\\star_\\sigma \\right)\\right),\n\\end{equation}\nwhere~$\\sigma$ is any permutation such that ${\\sf P}(\\sigma) = J$. Note that\nthe number of terms occurring in a coproduct $\\Delta\\left({\\bf P}_J\\right)$\nonly depends on the number~$n$ of nodes of each binary trees of~$J$ and\nis~$n + 1$. For example,\n\\begin{equation} \\begin{split}\n    \\Delta \\left(\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}} \\right)\n    & =\n    1\n    \\otimes\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}} \\\\\n    & +\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    \\otimes\n    1.\n\\end{split} \\end{equation}\n\\medskip\n\nFollowing Fomin~\\cite{Fom94} (see also~\\cite{BLL08}), we can build a\n\\emph{pair of graded graphs in duality} $(G_{\\bf P}, G_{{\\bf P}^\\star})$. The set\nof vertices of~$G_{\\bf P}$ and~$G_{{\\bf P}^\\star}$ is the set of pairs of twin\nbinary trees. There is an edge between the vertices~$J$ and~$J'$ in~$G_{\\bf P}$\n(resp. in~$G_{{\\bf P}^\\star}$) if~${\\bf P}_{J'}$ (resp.~${\\bf P}^\\star_{J'}$) appears\nin the product ${\\bf P}_J \\cdot {\\bf P}_{\n\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}$\n(resp. in the product ${\\bf P}^\\star_J \\cdot {\\bf P}^\\star_{\n\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}$).\nFigure~\\ref{fig:GrapheDualiteG} (resp. Figure~\\ref{fig:GrapheDualiteD})\nshows the graded graph~$G_{\\bf P}$ (resp.~$G_{{\\bf P}^\\star}$) restricted to vertices\nof order smaller than~$5$.\n\n\\begin{figure}[p]\n    \\scalebox{.145}{\n    \\begin{tikzpicture}\n       \n        \\scalebox{4}{\\node[minimum size = 6em](ee) at (0,0){$\\perp$ $\\perp$};}\n        \\node[fit=(ee)] (ee) {};\n       \n        \\node[Noeud](1g0)at(10,0){};\n        \\node[Noeud](1d0)at(11.5,0){};\n        \\node[fit=(1g0) (1d0)] (1) {};\n       \n        \\node[Noeud](12g0)at(20,20){};\n        \\node[Noeud](12g1)at(21,19){};\n        \\draw[Arete](12g0)--(12g1);\n        \\node[Noeud](12d0)at(22.5,19){};\n        \\node[Noeud](12d1)at(23.5,20){};\n        \\draw[Arete](12d1)--(12d0);\n        \\node[fit=(12g0) (12g1) (12d0) (12d1)] (12) {};\n       \n        \\node[Noeud](21g0)at(20,-21){};\n        \\node[Noeud](21g1)at(21,-20){};\n        \\draw[Arete](21g1)--(21g0);\n        \\node[Noeud](21d0)at(22.5,-20){};\n        \\node[Noeud](21d1)at(23.5,-21){};\n        \\draw[Arete](21d0)--(21d1);\n        \\node[fit=(21g0) (21g1) (21d0) (21d1)] (21) {};\n       \n        \\node[Noeud](123g0)at(40,50){};\n        \\node[Noeud](123g1)at(41,49){};\n        \\node[Noeud](123g2)at(42,48){};\n        \\draw[Arete](123g1)--(123g2);\n        \\draw[Arete](123g0)--(123g1);\n        \\node[Noeud](123d0)at(43.5,48){};\n        \\node[Noeud](123d1)at(44.5,49){};\n        \\draw[Arete](123d1)--(123d0);\n        \\node[Noeud](123d2)at(45.5,50){};\n        \\draw[Arete](123d2)--(123d1);\n        \\node[fit=(123g0) (123g1) (123g2) (123d0) (123d1) (123d2)] (123) {};\n       \n        \\node[Noeud](132g0)at(40,30){};\n        \\node[Noeud](132g1)at(41,28){};\n        \\node[Noeud](132g2)at(42,29){};\n        \\draw[Arete](132g2)--(132g1);\n        \\draw[Arete](132g0)--(132g2);\n        \\node[Noeud](132d0)at(43.5,29){};\n        \\node[Noeud](132d1)at(44.5,30){};\n        \\draw[Arete](132d1)--(132d0);\n        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(123)--(1243);\n        \\draw[Arete,line width=3.5pt] (123)--(1423);\n        \\draw[Arete,line width=3.5pt] (123)--(4123);\n        %\n        \\draw[Arete,line width=3.5pt] (132)--(1324);\n        \\draw[Arete,line width=3.5pt] (132)--(1342);\n        \\draw[Arete,line width=3.5pt] (132)--(1432);\n        \\draw[Arete,line width=3.5pt] (132)--(4132);\n        %\n        \\draw[Arete,line width=3.5pt] (312)--(3124);\n        \\draw[Arete,line width=3.5pt] (312)--(3142);\n        \\draw[Arete,line width=3.5pt] (312)--(4312);\n        %\n        \\draw[Arete,line width=3.5pt] (213)--(2134);\n        \\draw[Arete,line width=3.5pt] (213)--(2143);\n        \\draw[Arete,line width=3.5pt] (213)--(4213);\n        %\n        \\draw[Arete,line width=3.5pt] (231)--(2314);\n        \\draw[Arete,line width=3.5pt] (231)--(2341);\n        \\draw[Arete,line width=3.5pt] (231)--(2431);\n        \\draw[Arete,line width=3.5pt] (231)--(4231);\n        %\n        \\draw[Arete,line width=3.5pt] (321)--(3214);\n        \\draw[Arete,line width=3.5pt] (321)--(3241);\n        \\draw[Arete,line width=3.5pt] (321)--(3421);\n        \\draw[Arete,line width=3.5pt] (321)--(4321);\n    \\end{tikzpicture}}\n    \\caption{The graded graph~$G_{\\bf P}$ restricted to vertices of order\n    smaller than~$5$.}\n    \\label{fig:GrapheDualiteG}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\scalebox{.145}{\n    \\begin{tikzpicture}\n       \n        \\scalebox{4}{\\node[minimum size = 6em](ee) at (0,0){$\\perp$ $\\perp$};}\n        \\node[fit=(ee)] (ee) {};\n       \n        \\node[Noeud](1g0)at(10,0){};\n        \\node[Noeud](1d0)at(11.5,0){};\n        \\node[fit=(1g0) (1d0)] (1) {};\n       \n        \\node[Noeud](12g0)at(20,20){};\n        \\node[Noeud](12g1)at(21,19){};\n        \\draw[Arete](12g0)--(12g1);\n        \\node[Noeud](12d0)at(22.5,19){};\n        \\node[Noeud](12d1)at(23.5,20){};\n        \\draw[Arete](12d1)--(12d0);\n        \\node[fit=(12g0) (12g1) (12d0) (12d1)] (12) {};\n      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width=3.5pt,shorten >=7mm] (213)--(3241g0);\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (213)--(3142g0);\n        %\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (231)--(2314g0);\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (231)--(3421g0);\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (231)--(3142g0);\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (231)--(2143g0);\n        %\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (321)--(3214g0);\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (321)--(4312g0);\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (321)--(4321g0);\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (321)--(4213g0);\n    \\end{tikzpicture}}\n    \\caption{The graded graph~$G_{{\\bf P}^\\star}$ restricted to vertices of\n    order smaller than~$5$.}\n    \\label{fig:GrapheDualiteD}\n\\end{figure}\n\n\\subsubsection{A boolean basis}\nWe shall call a basis of an algebra (resp. coalgebra) a \\emph{boolean algebra basis}\n(resp. \\emph{boolean coalgebra basis}) if each element of the basis (resp.\ntensor square of the basis) only occurs with coefficient~$0$ or~$1$ in any\nproduct (resp. coproduct) involving two (resp. one) elements of the basis.\n\n\\begin{Proposition} \\label{prop:BaseEnsemblisteSSFQSym}\n    If~$\\equiv$ is an equivalence relation defined on~$A^*$ satisfying the\n    conditions of Theorem~\\ref{thm:HivertJanvier} and additionally, for\n    all~$\\pi, \\mu \\in \\mathfrak{S}$,\n    \\begin{equation} \\label{eq:BaseBool}\n        \\sigma, \\nu \\in \\pi \\cshuffle \\mu \\enspace\n        \\mbox{ and }\n        \\enspace \\sigma^{-1} \\equiv \\nu^{-1}\n        \\quad \\mbox{ imply } \\quad\n        \\sigma = \\nu,\n    \\end{equation}\n    then, the family\n    $\\left\\{{\\bf P}_{\\widehat{\\sigma}}\\right\\}_{\\widehat{\\sigma} \\in \\mathfrak{S}\/_\\equiv}$\n    defined in~(\\ref{eq:EquivFQSym}) is both an algebra and a coalgebra\n    boolean basis of the corresponding Hopf subalgebra of~${\\bf FQSym}$.\n\\end{Proposition}\n\\begin{proof}\n    It is immediate from the definition of the product of~${\\bf FQSym}$ that\n    $\\left\\{{\\bf P}_{\\widehat{\\sigma}}\\right\\}_{\\widehat{\\sigma} \\in \\mathfrak{S}\/_\\equiv}$\n    is a boolean algebra basis, regardless of~(\\ref{eq:BaseBool}).\n\n    By duality,\n    $\\left\\{{\\bf P}_{\\widehat{\\sigma}}\\right\\}_{\\widehat{\\sigma} \\in \\mathfrak{S}\/_\\equiv}$\n    is a boolean coalgebra basis if and only if its dual basis $\\left\\{{\\bf P}_{\\widehat{\\sigma}}^\\star\\right\\}_{\\widehat{\\sigma} \\in \\mathfrak{S}\/_\\equiv}$\n    is a boolean algebra basis. One has\n    \\begin{align}\n        {\\bf P}_{\\widehat{\\pi}}^\\star \\cdot {\\bf P}_{\\widehat{\\mu}}^\\star & =\n        \\phi\\left({\\bf F}_\\pi^\\star \\cdot {\\bf F}_\\mu^\\star\\right) \\\\\n        & = \\phi\\left(\\psi\\left(\\psi^{-1}\\left({\\bf F}_\\pi^\\star\\right) \\cdot \\psi^{-1}\\left({\\bf F}_\\mu^\\star\\right)\\right)\\right) \\\\\n        & = \\phi\\left(\\psi\\left({\\bf F}_{\\pi^{-1}} \\cdot {\\bf F}_{\\mu^{-1}}\\right)\\right) \\\\\n        & = \\sum_{\\sigma \\; \\in \\; \\pi^{-1} \\; \\cshuffle \\; \\nu^{-1}} \\phi\\left({\\bf F}_{\\sigma^{-1}}^\\star\\right) \\label{eq:PreuveBaseBool},\n    \\end{align}\n    where~$\\phi$ is the canonical projection mapping~${\\bf F}^\\star_\\sigma$\n    on~${\\bf P}^\\star_{\\widehat{\\sigma}}$ for any permutation~$\\sigma$, $\\psi$\n    is the Hopf isomorphism mapping~${\\bf F}_\\sigma$ on~${\\bf F}^\\star_{\\sigma^{-1}}$\n    for any permutation~$\\sigma$, and $\\pi \\in \\widehat{\\pi}$ and\n    $\\mu \\in \\widehat{\\mu}$. One can easily see that if~$\\equiv$ satisfies\n    the hypothesis of the proposition, then there are no multiplicities\n    in~(\\ref{eq:PreuveBaseBool}).\n\\end{proof}\n\nLaw and Reading have proved in~\\cite{LR10} that the basis of their Baxter\nHopf algebra, analog to our basis~$\\left\\{{\\bf P}_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$,\nis both a boolean algebra basis and a boolean coalgebra basis. We re-prove\nthis result in our setting:\n\\begin{Proposition} \\label{prop:BaseEnsembliste}\n    The basis~$\\left\\{{\\bf P}_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$ is both a boolean\n    algebra basis and a boolean coalgebra basis of~${\\bf Baxter}$.\n\\end{Proposition}\n\\begin{proof}\n    Let us prove that the sylvester equivalence relation satisfies the\n    assumptions of Proposition~\\ref{prop:BaseEnsemblisteSSFQSym}. Indeed,\n    the result directly follows from the fact that, by\n    Proposition~\\ref{prop:LienSylv}, the Baxter equivalence relation is\n    finer than the sylvester equivalence relation.\n\n    Let us start with a useful result: Let~$x$ and~$y$ be two words without\n    repetition of same length and $u, v \\in x \\cshuffle y$ (here, the letters\n    of~$y$ are shifted by~$\\max(x)$). Let us prove by induction on~$|x| + |y|$\n    that if~$\\operatorname{decr}(u)$ and~$\\operatorname{decr}(v)$ have same shape, then~$u = v$. It\n    is obvious if $|x| + |y| = 0$. Otherwise, one has $u = u' \\, {\\tt b} \\, u''$\n    and $v = v' \\, {\\tt b} \\, v''$ where ${\\tt b} := \\max(u) = \\max(v)$. Since the\n    shape of the left subtree of~$\\operatorname{decr}(u)$ is equal to the shape of the\n    left subtree of~$\\operatorname{decr}(v)$, the position of~${\\tt b}$ in~$u$ and~$v$ is\n    the same. Moreover, the word~$y$ is of the form $y = y' \\, {\\tt a} \\, y''$\n    where ${\\tt a} := \\max(y)$, and~$x$ is of the form~$x = x' x''$, where\n    $u', v' \\in x' \\cshuffle y'$ and $u'', v'' \\in x'' \\cshuffle y''$.\n    Since the left (resp. right) subtree of~$\\operatorname{decr}(u)$ is equal to the left\n    (resp. right) subtree of~$\\operatorname{decr}(v)$, by induction hypothesis,~$u' = v'$\n    and~$u'' = v''$, showing that~$u = v$.\n\n    Now, let $\\pi, \\mu \\in \\mathfrak{S}$ and $\\sigma \\ne \\nu \\in \\pi \\cshuffle \\mu$\n    and assume that $\\sigma^{-1} {\\:\\equiv_{\\operatorname{S}}\\:} \\nu^{-1}$. Then, by\n    Theorem~\\ref{thm:PSymbPBT}, the permutations~$\\sigma^{-1}$ and~$\\nu^{-1}$\n    give the same right binary search tree when inserted from right to left.\n    By Lemma~\\ref{lem:FormeInsIncr}, that implies that~$\\operatorname{decr}(\\sigma)$\n    and~$\\operatorname{decr}(\\nu)$ have same shape. That implies~$\\sigma = \\nu$,\n    contradicting our hypothesis.\n\\end{proof}\n\nBy duality, Proposition~\\ref{prop:BaseEnsembliste} also shows that the\nbasis $\\left\\{{\\bf P}_J^\\star\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$ is a boolean algebra\nand coalgebra basis.\n\n\\subsubsection{A lattice interval description of the product}\nIf~$\\equiv$ is an equivalence relation of~$\\mathfrak{S}$ and~$\\sigma$ a permutation,\ndenote by~$\\widehat{\\sigma} {\\!\\uparrow}$ (resp.~$\\widehat{\\sigma} {\\!\\downarrow}$)\nthe minimal (resp. maximal) permutation of the~$\\equiv$-equivalence class\nof~$\\sigma$ for the permutohedron order.\n\n\\begin{Proposition} \\label{prop:ProduitIntervalle}\n    If~$\\equiv$ is an equivalence relation defined on~$A^*$ satisfying the\n    conditions of Theorem~\\ref{thm:HivertJanvier} and additionally, the\n    $\\equiv$-equivalence classes of permutations are intervals of the\n    permutohedron, then the product on the family defined in~(\\ref{eq:EquivFQSym})\n    can be expressed as:\n    \\begin{equation}\n        {\\bf P}_{\\widehat{\\sigma}} \\cdot {\\bf P}_{\\widehat{\\nu}} =\n        \\sum_{\\substack{\\widehat{\\sigma} {\\!\\uparrow} {\\,\\diagup\\,} \\widehat{\\nu} {\\!\\uparrow}\n            {\\:\\leq_{\\operatorname{P}}\\:} \\pi {\\:\\leq_{\\operatorname{P}}\\:}\n            \\widehat{\\sigma} {\\!\\downarrow} {\\,\\diagdown\\,} \\widehat{\\nu} {\\!\\downarrow} \\\\\n            \\pi = \\min \\widehat{\\pi}}}\n        {\\bf P}_{\\widehat{\\pi}}.\n    \\end{equation}\n\\end{Proposition}\n\\begin{proof}\n    It is well-known that the shifted shuffle product of two permutohedron\n    intervals is still a permutohedron interval. Restating this fact\n    in~${\\bf FQSym}$, we have\n    \\begin{equation} \\label{eq:ProduitIntervalleFQSym}\n        \\left(\\sum_{\\sigma {\\:\\leq_{\\operatorname{P}}\\:} \\mu {\\:\\leq_{\\operatorname{P}}\\:} \\sigma'} {\\bf F}_\\mu \\right) \\cdot\n        \\left(\\sum_{\\nu {\\:\\leq_{\\operatorname{P}}\\:} \\tau {\\:\\leq_{\\operatorname{P}}\\:} \\nu'} {\\bf F}_\\tau \\right) =\n        \\sum_{\\sigma {\\,\\diagup\\,} \\nu {\\:\\leq_{\\operatorname{P}}\\:} \\pi {\\:\\leq_{\\operatorname{P}}\\:} \\sigma' {\\,\\diagdown\\,} \\nu'} {\\bf F}_\\pi.\n    \\end{equation}\n    By~(\\ref{eq:ProduitIntervalleFQSym}) and since that every $\\equiv$-equivalence\n    class is an interval of the permutohedron, we obtain\n    \\begin{equation} \\label{eq:ExpressionPP}\n        {\\bf P}_{\\widehat{\\sigma}} \\cdot {\\bf P}_{\\widehat{\\nu}} =\n            \\sum_{\\widehat{\\sigma} {\\!\\uparrow} {\\,\\diagup\\,} \\widehat{\\nu} {\\!\\uparrow}\n            {\\:\\leq_{\\operatorname{P}}\\:} \\pi {\\:\\leq_{\\operatorname{P}}\\:}\n            \\widehat{\\sigma} {\\!\\downarrow} {\\,\\diagdown\\,} \\widehat{\\nu} {\\!\\downarrow}} {\\bf F}_\\pi.\n    \\end{equation}\n    By Theorem~\\ref{thm:HivertJanvier}, the expression~(\\ref{eq:ExpressionPP})\n    can be expressed as a sum of~${\\bf P}_{\\widehat{\\pi}}$ elements and the\n    proposition follows.\n\\end{proof}\n\nLet $J_0 := (T^0_L, T^0_R)$ and $J_1 := (T^1_L, T^1_R)$ be two pairs of\ntwin binary trees. Let us define the pair of twin binary trees~$J_0 {\\,\\diagup\\,} J_1$\nby\n\\begin{equation}\n    J_0 {\\,\\diagup\\,} J_1 := (T^0_L {\\,\\diagdown\\,} T^1_L, \\; T^0_R {\\,\\diagup\\,} T^1_R).\n\\end{equation}\nIn the same way, the pair of twin binary trees~$J_0 {\\,\\diagdown\\,} J_1$ is defined\nby\n\\begin{equation}\n    J_0 {\\,\\diagdown\\,} J_1 := (T^0_L {\\,\\diagup\\,} T^1_L, \\; T^0_R {\\,\\diagdown\\,} T^1_R).\n\\end{equation}\n\nProposition~\\ref{prop:ProduitIntervalle} leads to the following expression\nfor the product of~${\\bf Baxter}$.\n\\begin{Corollaire}\n    For all pairs of twin binary trees~$J_0$ and~$J_1$, the product\n    of~${\\bf Baxter}$ satisfies\n    \\begin{equation} \\label{eq:ProdBXInter}\n        {\\bf P}_{J_0} \\cdot {\\bf P}_{J_1} =\n        \\sum_{J_0 {\\,\\diagup\\,} J_1 {\\:\\leq_{\\operatorname{B}}\\:} J {\\:\\leq_{\\operatorname{B}}\\:} J_0 {\\,\\diagdown\\,} J_1} {\\bf P}_J.\n    \\end{equation}\n\\end{Corollaire}\n\\begin{proof}\n    Let~$\\sigma$ and~$\\nu$ two permutations. It is immediate, from the\n    definition of the ${\\sf P}$-symbol algorithm, that the ${\\sf P}$-symbol\n    of the permutation~$\\sigma {\\,\\diagup\\,} \\nu$ (resp.~$\\sigma {\\,\\diagdown\\,} \\nu$) is\n    the pair of twin binary trees ${\\sf P}(\\sigma) {\\,\\diagup\\,} {\\sf P}(\\nu)$\n    (resp. ${\\sf P}(\\sigma) {\\,\\diagdown\\,} {\\sf P}(\\nu)$). The expression~(\\ref{eq:ProdBXInter})\n    follows from the fact that ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes of permutations\n    are intervals of the permutohedron (Proposition~\\ref{prop:EquivBXInter})\n    and from Proposition~\\ref{prop:ProduitIntervalle}.\n\\end{proof}\n\n\\subsubsection{Multiplicative bases and free generators}\nRecall that the \\emph{elementary} family\n$\\left\\{{\\bf E}^\\sigma\\right\\}_{\\sigma \\in \\mathfrak{S}}$ and the \\emph{homogeneous}\nfamily $\\left\\{{\\bf H}^\\sigma\\right\\}_{\\sigma \\in \\mathfrak{S}}$ of~${\\bf FQSym}$\nrespectively defined by\n\\begin{align}\n    {\\bf E}^\\sigma  & := \\sum_{\\sigma {\\:\\leq_{\\operatorname{P}}\\:} \\sigma'} {\\bf F}_{\\sigma'}, \\\\[.5em]\n    {\\bf H}^\\sigma & := \\sum_{\\sigma' {\\:\\leq_{\\operatorname{P}}\\:} \\sigma} {\\bf F}_{\\sigma'},\n\\end{align}\nform multiplicative bases of~${\\bf FQSym}$ (see~\\cite{AS05,DHNT11} for an exposition\nof some known bases of~${\\bf FQSym}$). Indeed, for all~$\\sigma, \\nu \\in \\mathfrak{S}$,\nthe product satisfies\n\\begin{align}\n    {\\bf E}^\\sigma \\cdot {\\bf E}^\\nu   & = {\\bf E}^{\\sigma {\\,\\diagup\\,} \\nu}, \\\\\n    {\\bf H}^\\sigma \\cdot {\\bf H}^\\nu & = {\\bf H}^{\\sigma {\\,\\diagdown\\,} \\nu}.\n\\end{align}\n\nMimicking these definitions, let us define the \\emph{elementary} family\n$\\left\\{{\\bf E}_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$ and the \\emph{homogeneous} family\n$\\left\\{{\\bf H}_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$ of ${\\bf Baxter}$ respectively by\n\\begin{align}\n    {\\bf E}_J  & := \\sum_{J {\\:\\leq_{\\operatorname{B}}\\:} J'} {\\bf P}_{J'}, \\\\[.5em]\n    {\\bf H}_J & := \\sum_{J' {\\:\\leq_{\\operatorname{B}}\\:} J} {\\bf P}_{J'}.\n\\end{align}\nThese families are bases of ${\\bf Baxter}$ since they are defined by triangularity.\n\n\\begin{Proposition} \\label{prop:BaseEHBaxter}\n    Let $J$ be a pair of twin binary trees and $\\sigma {\\!\\uparrow}$ (resp.\n    $\\sigma {\\!\\downarrow}$) be the minimal (resp. maximal) permutation such that\n    ${\\sf P}(\\sigma {\\!\\uparrow}) = J$ (resp. ${\\sf P}(\\sigma {\\!\\downarrow}) = J$). Then,\n    \\begin{align}\n        {\\bf E}_J  & = {\\bf E}^{\\sigma {\\!\\uparrow}}, \\\\\n        {\\bf H}_J & = {\\bf H}^{\\sigma {\\!\\downarrow}}.\n    \\end{align}\n\\end{Proposition}\n\\begin{proof}\n    Using the fact that, by Theorem~\\ref{thm:OrdreBaxterMinMax},\n    the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence relation is a lattice congruence of the\n    permutohedron, one successively has\n    \\begin{equation}\n        {\\bf E}_J = \\sum_{J {\\:\\leq_{\\operatorname{B}}\\:} J'} {\\bf P}_{J'}\n             = \\sum_{J {\\:\\leq_{\\operatorname{B}}\\:} J'}\n             \\sum_{\\substack{\\nu \\; \\in \\; \\mathfrak{S} \\\\ {\\sf P}(\\nu) = J'}} {\\bf F}_\\nu\n             = \\sum_{\\substack{\\nu \\; \\in \\; \\mathfrak{S} \\\\ J {\\:\\leq_{\\operatorname{B}}\\:} {\\sf P}(\\nu)}}\n             {\\bf F}_\\nu\n             = \\sum_{\\substack{\\nu \\; \\in \\; \\mathfrak{S} \\\\\n             \\sigma {\\!\\uparrow} {\\:\\leq_{\\operatorname{P}}\\:} \\nu}} {\\bf F}_\\nu\n            = {\\bf E}^{\\sigma {\\!\\uparrow}}.\n    \\end{equation}\n    The proof for the homogeneous family is analogous.\n\\end{proof}\n\n\\begin{Corollaire} \\label{cor:BasesMult}\n    For all pairs of twin binary trees $J_0$ and $J_1$, we have\n    \\begin{align}\n        {\\bf E}_{J_0} \\cdot {\\bf E}_{J_1}   & = {\\bf E}_{J_0 {\\,\\diagup\\,} J_1}, \\\\[.5em]\n        {\\bf H}_{J_0} \\cdot {\\bf H}_{J_1} & = {\\bf H}_{J_0 {\\,\\diagdown\\,} J_1}.\n    \\end{align}\n\\end{Corollaire}\n\\begin{proof}\n    Let~$\\sigma$ and~$\\nu$ be the minimal permutations of the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\n    classes respectively encoded by~$J_0$ and~$J_1$. By\n    Proposition~\\ref{prop:BaseEHBaxter}, we have\n    \\begin{equation}\n        {\\bf E}_{J_0} \\cdot {\\bf E}_{J_1} = {\\bf E}^\\sigma \\cdot {\\bf E}^\\nu = {\\bf E}^{\\sigma {\\,\\diagup\\,} \\nu}.\n    \\end{equation}\n    The permutation $\\sigma {\\,\\diagup\\,} \\nu$ is obviously the minimal element\n    of its ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class, and, by the definition of the\n    ${\\sf P}$-symbol algorithm, the ${\\sf P}$-symbol of $\\sigma {\\,\\diagup\\,} \\nu$ is\n    the pair of twin binary trees ${\\sf P}(\\sigma) {\\,\\diagup\\,} {\\sf P}(\\nu) = J_0 {\\,\\diagup\\,} J_1$.\n    The proof of the second part of the proposition is analogous.\n\\end{proof}\n\nFor example,\n\\begin{align}\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](4)--(2);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}}\n    \\cdot\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,0){};\n        \\node[Noeud,Marque1](1)at(1,-2){};\n        \\node[Noeud,Marque1](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete,Marque1](1)--(0);\n        \\node[Noeud,Marque1](2)at(2,-2){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud,Marque1](3)at(3,0){};\n        \\draw[Arete](3)--(1);\n    \\end{tikzpicture}}}\n    & =\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-1){};\n        \\node[Noeud,Marque1](5)at(5,-2){};\n        \\node[Noeud,Marque1](6)at(6,-4){};\n        \\node[Noeud,Marque1](7)at(7,-3){};\n        \\draw[Arete](7)--(6);\n        \\node[Noeud,Marque1](8)at(8,-4){};\n        \\draw[Arete](7)--(8);\n        \\draw[Arete](5)--(7);\n        \\draw[Arete](4)--(5);\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-4){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-5){};\n        \\node[Noeud](3)at(3,-6){};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud](4)at(4,-4){};\n        \\draw[Arete](4)--(2);\n        \\draw[Arete](1)--(4);\n        \\node[Noeud,Marque1](5)at(5,-2){};\n        \\draw[Arete](5)--(1);\n        \\node[Noeud,Marque1](6)at(6,-1){};\n        \\draw[Arete](6)--(5);\n        \\node[Noeud,Marque1](7)at(7,-2){};\n        \\draw[Arete](6)--(7);\n        \\node[Noeud,Marque1](8)at(8,0){};\n        \\draw[Arete](8)--(6);\n    \\end{tikzpicture}}}, \\\\[1em]\n    {\\bf H}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](4)--(2);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}}\n    \\cdot\n    {\\bf H}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,0){};\n        \\node[Noeud,Marque1](1)at(1,-2){};\n        \\node[Noeud,Marque1](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete,Marque1](1)--(0);\n        \\node[Noeud,Marque1](2)at(2,-2){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud,Marque1](3)at(3,0){};\n        \\draw[Arete](3)--(1);\n    \\end{tikzpicture}}}\n    & =\n    {\\bf H}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-4){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\node[Noeud,Marque1](5)at(5,0){};\n        \\draw[Arete](5)--(3);\n        \\node[Noeud,Marque1](6)at(6,-2){};\n        \\node[Noeud,Marque1](7)at(7,-1){};\n        \\draw[Arete](7)--(6);\n        \\node[Noeud,Marque1](8)at(8,-2){};\n        \\draw[Arete](7)--(8);\n        \\draw[Arete](5)--(7);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](4)--(2);\n        \\node[Noeud,Marque1](5)at(5,-4){};\n        \\node[Noeud,Marque1](6)at(6,-3){};\n        \\draw[Arete](6)--(5);\n        \\node[Noeud,Marque1](7)at(7,-4){};\n        \\draw[Arete](6)--(7);\n        \\node[Noeud,Marque1](8)at(8,-2){};\n        \\draw[Arete](8)--(6);\n        \\draw[Arete](4)--(8);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}}.\n\\end{align}\n\\medskip\n\nCorollary~\\ref{cor:BasesMult} also shows that the $\\left\\{{\\bf E}_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$\nand $\\left\\{{\\bf H}_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$ bases of~${\\bf Baxter}$ are boolean\nalgebra bases. However, these are not boolean coalgebra bases since one has\n\\begin{equation}\n    \\Delta \\left({\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n        \\node[Noeud](1)at(1.0,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}} \\right) =\n    1 \\otimes\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n        \\node[Noeud](1)at(1.0,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}}\n    + 2\\,\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n    \\end{tikzpicture}}}\n    +\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n        \\node[Noeud](1)at(1.0,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}}\n    \\otimes 1,\n\\end{equation}\nand\n\\begin{equation}\n    \\Delta \\left({\\bf H}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n        \\node[Noeud](1)at(1.0,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}} \\right) =\n    1 \\otimes\n    {\\bf H}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n        \\node[Noeud](1)at(1.0,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}}\n    + 2\\,\n    {\\bf H}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf H}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n    \\end{tikzpicture}}}\n    +\n    {\\bf H}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n        \\node[Noeud](1)at(1.0,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}}\n    \\otimes 1.\n\\end{equation}\n\\medskip\n\nLet us say that a pair of twin binary trees~$J$ is \\emph{connected} (resp.\n\\emph{anti-connected}) if all the permutations~$\\sigma$ such that\n${\\sf P}(\\sigma) = J$ are connected (resp. anti-connected). Since for any\nconnected (resp. anti-connected) permutation~$\\sigma$ and a permutation~$\\nu$\nsuch that~$\\sigma {\\:\\leq_{\\operatorname{P}}\\:} \\nu$ (resp.~$\\nu {\\:\\leq_{\\operatorname{P}}\\:} \\sigma$) the\npermutation~$\\nu$ is also connected (resp. anti-connected), it is enough\nto check if the minimal (resp. maximal) permutation of the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\nclass encoded by~$J$ is connected (resp. anti-connected) to decide if~$J$\nis connected (resp. anti-connected).\n\n\\begin{Lemme} \\label{lem:ABJConnexes}\n    For any pair of twin binary trees~$J$, there exists a sequence of\n    connected (resp. anti-connected) pairs of twin binary trees\n    $J_1$, \\dots, $J_k$ such that\n    \\begin{equation}\n        J = J_1 {\\,\\diagup\\,} \\cdots {\\,\\diagup\\,} J_k \\quad\n        \\mbox{(resp. $J = J_1 {\\,\\diagdown\\,} \\cdots {\\,\\diagdown\\,} J_k$)}.\n    \\end{equation}\n\\end{Lemme}\n\\begin{proof}\n    Let~$\\sigma$ be the minimal permutation of the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\n    class encoded by $J$ (recall that the existence of this element is ensured\n    by Proposition~\\ref{prop:EquivBXInter}). One can write~$\\sigma$ as\n    \\begin{equation}\n        \\sigma = \\sigma^{(1)} {\\,\\diagup\\,} \\cdots {\\,\\diagup\\,} \\sigma^{(k)},\n    \\end{equation}\n    where the permutations~$\\sigma^{(i)}$ are connected for all $1 \\leq i \\leq k$.\n    Since~$\\sigma$ is the minimal permutation of its ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\n    class, all the permutations~$\\sigma^{(i)}$ are also minimal of their\n    ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes. Hence, the pairs of twin binary trees\n    ${\\sf P}\\left(\\sigma^{(i)}\\right)$ are connected and we can write\n    \\begin{equation}\n        J = {\\sf P}\\left(\\sigma^{(1)}\\right)\n        {\\,\\diagup\\,} \\cdots {\\,\\diagup\\,}\n        {\\sf P}\\left(\\sigma^{(k)}\\right).\n    \\end{equation}\n    The proof for the respective part is analogous.\n\\end{proof}\n\n\\begin{Theoreme} \\label{thm:LiberteBaxter}\n    The algebra~${\\bf Baxter}$ is free on the elements~${\\bf E}_J$ (resp.~${\\bf H}_J$)\n    such that~$J$ is a connected (resp. anti-connected) pair of twin\n    binary trees.\n\\end{Theoreme}\n\\begin{proof}\n    By Corollary~\\ref{cor:BasesMult} and Lemma~\\ref{lem:ABJConnexes},\n    each element~${\\bf E}_J$ can be expressed as\n    \\begin{equation}\n        {\\bf E}_J = {\\bf E}_{J_1} \\cdot \\dots \\cdot {\\bf E}_{J_k},\n    \\end{equation}\n    where the pairs of twin binary trees~$J_i$ are connected for all\n    $1 \\leq i \\leq k$.\n\n    Now, since for all permutations $\\sigma$ and $\\nu$ one has\n    ${\\bf E}^\\sigma \\cdot {\\bf E}^\\nu = {\\bf E}^{\\sigma {\\,\\diagup\\,} \\nu}$ in ${\\bf FQSym}$, and\n    since any permutation $\\sigma$ admits a unique expression\n    \\begin{equation}\n        \\sigma = \\sigma^{(1)} {\\,\\diagup\\,} \\cdots {\\,\\diagup\\,} \\sigma^{(k)},\n    \\end{equation}\n    where~$\\sigma^{(1)}$, \\dots, $\\sigma^{(k)}$ are connected permutations,\n    there is no relation in~${\\bf FQSym}$ between the elements~${\\bf E}^\\sigma$\n    where~$\\sigma$ is a connected permutation.\n\n    Hence, by Proposition~\\ref{prop:BaseEHBaxter} and Corollary~\\ref{cor:BasesMult},\n    there is also no relation in~${\\bf Baxter}$ between the elements~${\\bf E}_J$\n    where~$J$ is a connected pair of twin binary trees. The proof for the\n    respective part is analogous.\n\\end{proof}\n\nLet us denote by~$B_C(z)$ the generating series of connected (resp. anti-connected)\npairs of twin binary trees. It follows, from Theorem~\\ref{thm:LiberteBaxter},\nthat the Hilbert series~$B(z)$ of~${\\bf Baxter}$ satisfies\n$B(z) = 1 \/ \\left(1 - B_C(z)\\right)$. Hence, the generating series~$B_C(z)$\nsatisfies\n\\begin{equation} \\label{eq:SGGenAlg}\n    B_C(z) = 1 - \\frac{1}{B(z)}.\n\\end{equation}\nFirst dimensions of algebraic generators of~${\\bf Baxter}$ are\n\\begin{equation}\n    0, 1, 1, 3, 11, 47, 221, 1113, 5903, 32607, 186143, 1092015.\n\\end{equation}\n\nHere follows algebraic generators of ${\\bf Baxter}$ of order $1$ to $4$:\n\\begin{equation}\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}};\n\\end{equation}\n\n\\begin{equation}\n{\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}};\n\\end{equation}\n\n\\begin{equation}\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](1)--(2);\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}};\n\\end{equation}\n\n\\begin{equation}\\begin{split}\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](0)--(3);\n    \\end{tikzpicture}}}, &\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(1);\n        \\draw[Arete](0)--(3);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\node[Noeud](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(0);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}, \\\\\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(0);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}, &\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(1);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(1);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}}.\n\\end{split}\\end{equation}\n\\medskip\n\n\\begin{Proposition}\n    If~$\\sigma$ is a connected (resp. anti-connected) Baxter permutation,\n    then any permutation~$\\nu$ such that~$\\sigma {\\:\\equiv_{\\operatorname{B}}\\:} \\nu$ is also\n    connected (resp. anti-connected).\n\\end{Proposition}\n\\begin{proof}\n    As any permutation, every Baxter permutation~$\\sigma$ can be uniquely\n    expressed as\n    \\begin{equation}\n        \\sigma = \\sigma^{(1)} {\\,\\diagup\\,} \\cdots {\\,\\diagup\\,} \\sigma^{(k)},\n    \\end{equation}\n    where the permutations~$\\sigma^{(i)}$ are connected for all $1 \\leq i \\leq k$.\n    Moreover, since~$\\sigma$ avoids the permutation patterns $2-41-3$\n    and $3-14-2$, the permutations~$\\sigma^{(i)}$ also does, and hence,\n    the~$\\sigma^{(i)}$ are Baxter permutations. This shows that the generating\n    series of connected Baxter permutations is~$B_C(z)$ and thus, that\n    connected Baxter permutations, connected pairs of twin binary trees,\n    and connected minimal permutations of Baxter equivalence classes are\n    equinumerous.\n\n    The proposition follows from Theorem~\\ref{thm:EquivBXBaxter} saying\n    that each ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class of permutations contains\n    exactly one Baxter permutation. The proof for the respective part is\n    analogous.\n\\end{proof}\n\n\\begin{Corollaire}\n    The algebra~${\\bf Baxter}$ is free on the elements~${\\bf E}_J$ (resp.~${\\bf H}_J$)\n    where the Baxter permutation belonging to the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\n    class encoded by~$J$ is connected (resp. anti-connected).\n\\end{Corollaire}\n\n\\subsubsection{Bidendriform bialgebra structure and self-duality}\nA Hopf algebra $(H, \\cdot, \\Delta)$ can be fit into a bidendriform\nbialgebra structure~\\cite{Foi07} if $(H^+, \\prec, \\succ)$ is a dendriform\nalgebra~\\cite{Lod01} and $(H^+, \\Delta_\\Gauche, \\Delta_\\Droite)$ a codendriform coalgebra,\nwhere~$H^+$ is the augmentation ideal of~$H$. The operators~$\\prec$, $\\succ$,\n$\\Delta_\\Gauche$ and~$\\Delta_\\Droite$ have to fulfill some compatibility relations. In\nparticular, for all~$x, y \\in H^+$, the product~$\\cdot$ of~$H$ is retrieved\nby $x \\cdot y = x \\prec y + x \\succ y$ and the coproduct~$\\Delta$ of~$H$\nis retrieved by $\\Delta(x) = 1 \\otimes x + \\Delta_\\Gauche(x) + \\Delta_\\Droite(x) + x \\otimes 1$.\nRecall that an element~$x \\in H^+$ is \\emph{totally primitive} if\n$\\Delta_\\Gauche(x) = 0 = \\Delta_\\Droite(x)$.\n\\medskip\n\nThe Hopf algebra~${\\bf FQSym}$ admits a bidendriform bialgebra structure~\\cite{Foi07}.\nIndeed, for all~$\\sigma, \\nu \\in \\mathfrak{S}_n$ with~$n \\geq 1$, set\n\\begin{equation}\n    {\\bf F}_\\sigma \\prec {\\bf F}_\\nu :=\n    \\sum_{\\substack{\\pi \\; \\in \\; \\sigma \\; \\cshuffle \\; \\nu \\\\\n    \\pi_{|\\pi|} = \\sigma_{|\\sigma|}}} {\\bf F}_\\pi,\n\\end{equation}\n\\begin{equation}\n    {\\bf F}_\\sigma \\succ {\\bf F}_\\nu :=\n    \\sum_{\\substack{\\pi \\; \\in \\; \\sigma \\; \\cshuffle \\; \\nu \\\\\n    \\pi_{|\\pi|} = \\nu_{|\\nu|} + |\\sigma|}} {\\bf F}_\\pi,\n\\end{equation}\n\\begin{equation}\n    \\Delta_\\Gauche({\\bf F}_\\sigma) :=\n    \\sum_{\\substack{\\sigma = uv \\\\ \\max(u) = \\max(\\sigma)}}\n    {\\bf F}_{\\operatorname{std}(u)} \\otimes {\\bf F}_{\\operatorname{std}(v)},\n\\end{equation}\n\\begin{equation}\n    \\Delta_\\Droite({\\bf F}_\\sigma) := \\sum_{\\substack{\\sigma = uv \\\\\n    \\max(v) = \\max(\\sigma)}}\n    {\\bf F}_{\\operatorname{std}(u)} \\otimes {\\bf F}_{\\operatorname{std}(v)}.\n\\end{equation}\n\n\\begin{Proposition} \\label{prop:MemeLettreEquiv}\n    If~$\\equiv$ is an equivalence relation defined on~$A^*$ satisfying the\n    conditions of Theorem~\\ref{thm:HivertJanvier} and additionally, for\n    all~$u, v \\in A^*$, the relation~$u \\equiv v$ implies $u_{|u|} = v_{|v|}$,\n    then, the family defined in~(\\ref{eq:EquivFQSym}) spans a bidendriform\n    sub-bialgebra of~${\\bf FQSym}$ that is free as an algebra, cofree as a coalgebra,\n    self-dual, free as a dendriform algebra on its totally primitive elements,\n    and the Lie algebra of its primitive elements is free.\n\\end{Proposition}\n\\begin{proof}\n    It is enough to show that the operators~$\\prec$, $\\succ$, $\\Delta_\\Gauche$\n    and~$\\Delta_\\Droite$ of~${\\bf FQSym}$ are well-defined in the Hopf subalgebra~$H$\n    of~${\\bf FQSym}$ spanned by the elements\n    $\\left\\{{\\bf P}_{\\widehat{\\sigma}}\\right\\}_{\\widehat{\\sigma} \\in \\mathfrak{S}\/_\\equiv}$.\n    In this way, $H$ is endowed with a structure of bidendriform bialgebra\n    and the results of Foissy~\\cite{Foi07} imply the rest of the proposition.\n\n    Fix $\\widehat{\\sigma}, \\widehat{\\nu} \\in \\mathfrak{S}\/_\\equiv$ and an\n    element~${\\bf F}_\\pi$ appearing in the product\n    ${\\bf P}_{\\widehat{\\sigma}} \\prec {\\bf P}_{\\widehat{\\nu}}$. Hence, there is\n    a permutation~$\\sigma \\in \\widehat{\\sigma}$ such that\n    $\\pi_{|\\pi|} = \\sigma_{|\\sigma|}$. Let~$\\pi'$ a permutation such\n    that~$\\pi \\equiv \\pi'$. By Theorem~\\ref{thm:HivertJanvier}, the\n    element~${\\bf F}_{\\pi'}$ appears in the product\n    ${\\bf P}_{\\widehat{\\sigma}} \\cdot {\\bf P}_{\\widehat{\\nu}}$, and hence, it also\n    appears in ${\\bf P}_{\\widehat{\\sigma}} \\prec {\\bf P}_{\\widehat{\\nu}}$\n    or in ${\\bf P}_{\\widehat{\\sigma}} \\succ {\\bf P}_{\\widehat{\\nu}}$. Assume by\n    contradiction that~${\\bf F}_{\\pi'}$ appears in\n    ${\\bf P}_{\\widehat{\\sigma}} \\succ {\\bf P}_{\\widehat{\\nu}}$.\n    There are two permutations~$\\sigma' \\in \\widehat{\\sigma}$\n    and~$\\nu' \\in \\widehat{\\nu}$ such that\n    $\\pi'_{|\\pi'|} = \\nu'_{|\\nu'|} + |\\sigma'|$. That implies that\n    $\\pi_{|\\pi|} \\ne \\pi'_{|\\pi'|}$ and contradicts the fact that all\n    permutations of a same $\\equiv$-equivalence class end with a same\n    letter. Hence, the element~${\\bf F}_{\\pi'}$ appears in\n    ${\\bf P}_{\\widehat{\\sigma}} \\prec {\\bf P}_{\\widehat{\\nu}}$, showing that\n    the product~$\\prec$ is well-defined in~$H$. Then so is~$\\succ$\n    since~$\\prec + \\succ$ is the whole product.\n\n    Fix $\\widehat{\\sigma} \\in \\mathfrak{S}\/_\\equiv$ and an\n    element ${\\bf F}_\\nu \\otimes {\\bf F}_\\pi$ appearing in the coproduct\n    $\\Delta_\\Gauche({\\bf P}_{\\widehat{\\sigma}})$. Hence, there is a\n    permutation~$\\sigma \\in \\widehat{\\sigma}$ such that~$\\sigma = uv$,\n    $\\nu = \\operatorname{std}(u)$, $\\pi = \\operatorname{std}(v)$ and the maximal letter of~$uv$ is\n    in the factor~$u$. Now, let~$\\nu'$ and~$\\pi'$ be two permutations such\n    that $\\nu \\equiv \\nu'$, $\\pi \\equiv \\pi'$. Let us show that the element\n    ${\\bf F}_{\\nu'} \\otimes {\\bf F}_{\\pi'}$ also appears in~$\\Delta_\\Gauche({\\bf P}_{\\widehat{\\sigma}})$.\n    For that, let~$u'$ be a permutation of~$u$ such that~$\\operatorname{std}(u') = \\nu'$,\n    and~$v'$ be a permutation of~$v$ such that~$\\operatorname{std}(v') = \\pi'$. Since\n    $\\operatorname{ev}(u') = \\operatorname{ev}(u)$, $\\operatorname{std}(u') \\equiv \\operatorname{std}(u)$, and~$\\equiv$ is\n    compatible with the destandardization process, one has~$u \\equiv u'$.\n    For the same reason,~$v \\equiv v'$, and since~$\\equiv$ is a congruence,\n    one has~$uv \\equiv u'v'$. Finally, since the maximal letter of~$uv$\n    is in~$u$, the maximal letter of~$u'v'$ is in~$u'$, showing that the\n    element ${\\bf F}_{\\nu'} \\otimes {\\bf F}_{\\pi'}$ appears in\n    $\\Delta_\\Gauche({\\bf P}_{\\widehat{\\sigma}})$. Thus, the coproduct~$\\Delta_\\Gauche$\n    is well-defined in~$H$. The proof for the coproduct~$\\Delta_\\Droite$ is analogous.\n\\end{proof}\n\n\\begin{Corollaire} \\label{cor:BaxterBidendr}\n    The Hopf algebra ${\\bf Baxter}$ is free as an algebra, cofree as a coalgebra,\n    self-dual, free as a dendriform algebra on its totally primitive elements,\n    and the Lie algebra of its primitive elements is free.\n\\end{Corollaire}\n\\begin{proof}\n    Since all words of a same ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class end with a same\n    letter,~${\\:\\equiv_{\\operatorname{B}}\\:}$ satisfies the premises of\n    Proposition~\\ref{prop:MemeLettreEquiv} and hence,~${\\bf Baxter}$ satisfies\n    all stated properties.\n\\end{proof}\n\nConsidering the map $\\theta' : {\\bf PBT} \\hookrightarrow {\\bf FQSym}$ that is the\ninjection from~${\\bf PBT}$ to~${\\bf FQSym}$ and\n$\\phi' : {\\bf FQSym}^\\star \\twoheadrightarrow {\\bf PBT}^\\star$ the surjection\nfrom~${\\bf FQSym}^\\star$ to ${\\bf PBT}^\\star$, it is well-known (see~\\cite{HNT05})\nthat the map $\\phi' \\circ \\psi \\circ \\theta'$ induces an isomorphism\nbetween~${\\bf PBT}$ and~${\\bf PBT}^\\star$. Hence, since by Corollary~\\ref{cor:BaxterBidendr},\nthe Hopf algebras~${\\bf Baxter}$ and~${\\bf Baxter}^\\star$ are isomorphic, it is natural\nto test if an analogous map is still an isomorphism between~${\\bf Baxter}$\nand~${\\bf Baxter}^\\star$. However, denoting by $\\theta : {\\bf Baxter} \\hookrightarrow {\\bf FQSym}$\nthe injection from~${\\bf Baxter}$ to~${\\bf FQSym}$, the map\n$\\phi \\circ \\psi \\circ \\theta : {\\bf Baxter} \\rightarrow {\\bf Baxter}^\\star$ is not\nan isomorphism. Indeed\n\\begin{align}\n    \\phi \\circ \\psi \\circ \\theta \\left(\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}} \\right)\n    & = \\phi \\circ \\psi \\left( {\\bf F}_{2143} + {\\bf F}_{2413} \\right)\n      = \\phi \\left( {\\bf F}^\\star_{2143} + {\\bf F}^\\star_{3142} \\right)\n      =\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}}, \\\\\n    \\phi \\circ \\psi \\circ \\theta \\left(\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}} \\right)\n    & = \\phi \\circ \\psi \\left( {\\bf F}_{3142} + {\\bf F}_{3412} \\right)\n      = \\phi \\left( {\\bf F}^\\star_{2413} + {\\bf F}^\\star_{3412} \\right)\n      =\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}},\n\\end{align}\nshowing that~$\\phi \\circ \\psi \\circ \\theta$ is not injective.\n\n\\subsubsection{Primitive and totally primitive elements}\n\nSince the family~$\\left\\{{\\bf E}_J\\right\\}_{J \\in C}$\n(resp.~$\\left\\{{\\bf H}_J\\right\\}_{J \\in C}$), where~$C$ is the set of connected\n(resp. anti-connected) pairs of twin binary trees are indecomposable elements\nof~${\\bf Baxter}$, its dual family~$\\left\\{{\\bf E}^\\star_J\\right\\}_{J \\in C}$\n(resp.~$\\left\\{{\\bf H}^\\star_J\\right\\}_{J \\in C}$) forms a basis of the Lie\nalgebra of  the primitive elements of~${\\bf Baxter}^\\star$. By\nCorollary~\\ref{cor:BaxterBidendr}, this Lie algebra is free.\n\\medskip\n\nFollowing~\\cite{Foi07}, the generating series $B_T(z)$ of the totally\nprimitive elements of ${\\bf Baxter}$ is\n\\begin{equation}\n    B_T(z) = \\frac{B(z) - 1}{B(z)^2}.\n\\end{equation}\nFirst dimensions of totally primitive  elements of ${\\bf Baxter}$ are\n\\begin{equation}\n    0, 1, 0, 1, 4, 19, 96, 511, 2832, 16215, 95374, 573837.\n\\end{equation}\n\nHere follows a basis of the totally primitive elements of ${\\bf Baxter}$ of\norder $1$, $3$ and $4$:\n\\begin{align}\n    t_{1, 1} & =\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}, \\\\[1em]\n    t_{3, 1} & =\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}}\n    -\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}, \\\\[1em]\n    t_{4, 1} & =\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\node[Noeud](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(1);\n        \\draw[Arete](0)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(1);\n        \\draw[Arete](0)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\node[Noeud](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}} \\\\\n    & -\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}}\n    -\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](0)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}\n    -\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}}, \\nonumber \\\\\n    t_{4, 2} & =\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}}\n    -\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](0)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}, \\\\\n    t_{4, 3} & =\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](0)--(3);\n    \\end{tikzpicture}}}\n    -\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}}, \\\\\n    t_{4, 4} & =\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    -\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}}.\n\\end{align}\n\n\\subsubsection{\\texorpdfstring{Compatibility with the $\\#$ product}\n                              {Compatibility with the sharp product}}\nAval and Viennot~\\cite{AV10} endowed~${\\bf PBT}$ with a new associative product\ncalled the \\emph{$\\#$ product}. The product of two elements of~${\\bf PBT}$\nof degrees~$n$ and~$m$ is an element of degree~$n + m - 1$. Aval, Novelli,\nand Thibon~\\cite{ANT11} generalized the $\\#$ product at the level of the\nassociative algebra and showed that it is still well-defined in~${\\bf FQSym}$.\n\\medskip\n\nLet for all~$k \\geq 1$ the linear maps $d_k : {\\bf FQSym} \\to {\\bf FQSym}$ defined\nfor any permutation~$\\sigma$ of~$\\mathfrak{S}_n$ by\n\\begin{equation}\n    d_k({\\bf F}_\\sigma) :=\n    \\begin{cases}\n        {\\bf F}_{\\operatorname{std}(\\sigma_1 \\dots \\sigma_i \\sigma_{i + 2} \\dots \\sigma_n)} &\n            \\mbox{if there is $1 \\leq i \\leq n - 1$ such that $\\sigma_i = k$\n            and $\\sigma_{i + 1} = k + 1$}, \\\\\n        0 & \\mbox{otherwise}.\n    \\end{cases}\n\\end{equation}\nNow, for any permutations~$\\sigma$ and~$\\nu$, the $\\#$-product is defined\nin~${\\bf FQSym}$ by\n\\begin{equation}\n    {\\bf F}_\\sigma \\# {\\bf F}_\\nu := d_n\\left({\\bf F}_\\sigma \\cdot {\\bf F}_\\nu\\right),\n\\end{equation}\nwhere~$n$ is the size of~$\\sigma$.\n\n\\begin{Proposition} \\label{prop:DkInterBaxter}\n    The linear maps~$d_k$ are well-defined in~${\\bf Baxter}$. More precisely,\n    one has for any pair of twin binary trees~$J := (T_0, T_1)$,\n    \\begin{equation}\n        d_k({\\bf P}_J) =\n        \\begin{cases}\n            {\\bf P}_{J'} &\n            \\substack{\\mbox{if the $k\\!+\\!1$-st (resp. $k$-th) node is a\n            child of the} \\\\\n            \\mbox{$k$-th (resp. $k\\!+\\!1$-st) node in $T_L$ (resp. $T_R$),}} \\\\\n            0 & \\mbox{otherwise},\n        \\end{cases}\n    \\end{equation}\n    where $J' := (T'_L, T'_R)$ is the pair of twin binary trees obtained\n    by contracting in~$T_L$ and~$T_R$ the edges connecting the\n    $k$-th and the $k\\!+\\!1$-st nodes.\n\\end{Proposition}\n\\begin{proof}\n    This proof relies on the fact that, according to Proposition~\\ref{prop:BXExtLin},\n    the permutations of a Baxter equivalence class coincide with linear\n    extensions of the posets~$\\bigtriangleup(T_L)$ and~$\\bigtriangledown(T_R)$.\n\n    We have two cases to consider whether the $k\\!+\\!1$-st (resp. $k$-th)\n    node is a child of the $k$-th (resp. $k\\!+\\!1$-st) node in~$T_L$\n    (resp.~$T_R$).\n    \\begin{enumerate}[label = {\\bf Case \\arabic*.}, fullwidth]\n        \\item If so, there is in the Baxter equivalence class represented\n        by~$J$ some permutations with a factor~$k.(k\\!+\\!1)$. The map~$d_k$\n        deletes letters~$k\\!+\\!1$ in these permutations and standardizes\n        them. The obtained permutations coincide with linear extensions\n        of the posets~$\\bigtriangleup(T'_L)$ and~$\\bigtriangledown(T'_R)$.\n        \\item If this is not the case, since the $k$-th and $k\\!+\\!1$-st nodes\n        of a binary tree are on a same path starting from the root, no\n        permutation of the Baxter class represented by~$J$ has a\n        factor~$k.(k\\!+\\!1)$. Hence,~$d_k({\\bf P}_J) = 0$. \\qedhere\n    \\end{enumerate}\n\\end{proof}\n\nOne has for example\n\\begin{equation}\n    d_3\\left({\\bf P}_{\n    \\scalebox{.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2.0,-2){};\n        \\node[Noeud](3)at(3.0,-3){};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud](4)at(4.0,-1){};\n        \\draw[Arete](4)--(2);\n        \\node[Noeud](5)at(5.0,-2){};\n        \\draw[Arete](4)--(5);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}} \\;\n    \\scalebox{.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-2){};\n        \\node[Noeud](1)at(1.0,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2.0,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3.0,0){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4.0,-2){};\n        \\node[Noeud](5)at(5.0,-1){};\n        \\draw[Arete](5)--(4);\n        \\draw[Arete](3)--(5);\n    \\end{tikzpicture}}} \\right) =\n    {\\bf P}_{\n    \\scalebox{.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2.0,-2){};\n        \\node[Noeud](3)at(3.0,-1){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4.0,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}} \\;\n    \\scalebox{.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2.0,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3.0,-2){};\n        \\node[Noeud](4)at(4.0,-1){};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n    \\end{tikzpicture}}}\\,.\n\\end{equation}\n\nProposition~\\ref{prop:DkInterBaxter} shows in particular that the $\\#$ product\nin well-defined in~${\\bf Baxter}$.\n\n\\subsection{\\texorpdfstring{Connections with other Hopf subalgebras of ${\\bf FQSym}$}\n                           {Connections with other Hopf subalgebras of FQSym}}\n\n\\subsubsection{\\texorpdfstring{Connection with the Hopf algebra ${\\bf PBT}$}\n                              {Connection with the Hopf algebra PBT}}\nWe already recalled that the sylvester congruence leads to the construction\nof the Hopf subalgebra~${\\bf PBT}$~\\cite{LR98} of~${\\bf FQSym}$, whose fundamental\nbasis\n\\begin{equation}\n    \\left\\{{\\bf P}_T : T \\in \\mathcal{B}\\mathcal{T}\\right\\}\n\\end{equation}\nis defined in accordance\nwith~(\\ref{eq:EquivFQSym}) (see~\\cite{HNT02} and~\\cite{HNT05}). By\nProposition~\\ref{prop:LienSylv}, every ${\\:\\equiv_{\\operatorname{S}}\\:}$-equivalence class is a\nunion of some ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes. Hence, we have the following\ninjective Hopf map:\n\\begin{equation}\n    \\rho : {\\bf PBT} \\hookrightarrow {\\bf Baxter},\n\\end{equation}\nsatisfying\n\\begin{equation}\n    \\rho \\left({\\bf P}_T\\right) =\n    \\sum_{\\substack{T' \\; \\in \\; \\mathcal{B}\\mathcal{T} \\\\ J := (T', T) \\; \\in \\; \\mathcal{T}\\mathcal{B}\\mathcal{T}}}\n    {\\bf P}_J,\n\\end{equation}\nfor any binary tree~$T$. For example,\n\\begin{align}\n    \\rho \\left(\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque1](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\node[Noeud,Marque1](4)at(4,-1){};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n    \\end{tikzpicture}}}\n    \\right)\n    & =\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(1);\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](0)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque1](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\node[Noeud,Marque1](4)at(4,-1){};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(0);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque1](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\node[Noeud,Marque1](4)at(4,-1){};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\node[Noeud](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-3){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(3);\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque1](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\node[Noeud,Marque1](4)at(4,-1){};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n    \\end{tikzpicture}}}.\n\\end{align}\n\n\\subsubsection{\\texorpdfstring{Connection with the Hopf algebra $\\DSym{3}$}\n                              {Connection with the Hopf algebra DSym3}}\nThe congruence~$\\EquivR{3}$ leads to the construction of the Hopf\nsubalgebra~$\\DSym{3}$ of~${\\bf FQSym}$, whose fundamental basis\n\\begin{equation}\n    \\left\\{{\\bf P}_{\\widehat{\\sigma}} :\n    \\widehat{\\sigma} \\in \\mathfrak{S} \/_{\\EquivR{3}}\\right\\}\n\\end{equation}\nis defined in accordance with~(\\ref{eq:EquivFQSym}) (see~\\cite{NRT11}).\nBy Proposition~\\ref{prop:Lien3Recul}, every $\\EquivR{3}$-equivalence\nclass of permutations is a union of some ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes.\nHence, we have the following injective Hopf map:\n\\begin{equation}\n    \\alpha : \\DSym{3} \\hookrightarrow {\\bf Baxter},\n\\end{equation}\nsatisfying\n\\begin{equation}\n    \\alpha \\left( {\\bf P}_{\\widehat{\\sigma}} \\right) =\n    \\sum_{\\sigma \\; \\in \\; \\widehat{\\sigma} \\cap \\mathfrak{S}^{\\operatorname{B}}} {\\bf P}_{{\\sf P}(\\sigma)},\n\\end{equation}\nfor any $\\EquivR{3}$-equivalence class~$\\widehat{\\sigma}$ of permutations.\n\n\\subsubsection{\\texorpdfstring{Connection with the Hopf algebra ${\\bf Sym}$}\n                              {Connection with the Hopf algebra Sym}}\nThe hypoplactic congruence~\\cite{N98} leads to the construction of the Hopf\nsubalgebra~${\\bf Sym}$ of~${\\bf FQSym}$. As already mentioned, the hypoplactic congruence\nis the same as the congruence~$\\EquivR{2}$ when both are restricted on\npermutations. Moreover, the hypoplactic equivalence classes of permutations\ncan  be encoded by binary words. Indeed, if~$\\widehat{\\sigma}$ is such an\nequivalence class,~$\\widehat{\\sigma}$ contains all the permutations having\na given recoil set. Thus, the class~$\\widehat{\\sigma}$ can be encoded by\nthe binary word~$b$ of length~$n - 1$ where~$n$ is the length of the elements\nof~$\\widehat{\\sigma}$ and~$b_i = 1$ if and only if~$i$ is a recoil of the elements\nof~$\\widehat{\\sigma}$. We denote by\n\\begin{equation}\n    \\left\\{{\\bf P}_b : b \\in \\{0, 1\\}^*\\right\\}\n\\end{equation}\nthe fundamental basis of~${\\bf Sym}$ indexed by binary words.\n\\medskip\n\nSince~${\\bf PBT}$ is a Hopf subalgebra of~${\\bf Baxter}$ and~${\\bf Sym}$ is a Hopf subalgebra\nof~${\\bf PBT}$~\\cite{HNT05}, ${\\bf Sym}$ is itself a Hopf subalgebra of~${\\bf Baxter}$.\nThe injective Hopf map\n\\begin{equation}\n    \\beta : {\\bf Sym} \\hookrightarrow {\\bf PBT},\n\\end{equation}\nsatisfies, thanks to the fact that the hypoplactic equivalence classes are\nunion of ${\\:\\equiv_{\\operatorname{S}}\\:}$-equivalence classes and Proposition~\\ref{prop:FeuillesInversions},\n\\begin{equation}\n    \\beta \\left({\\bf P}_b\\right) =\n    \\sum_{\\substack{T \\; \\in \\; \\mathcal{B}\\mathcal{T} \\\\ \\operatorname{cnp}(T) = b}} {\\bf P}_T,\n\\end{equation}\nfor any binary word~$b$. From a combinatorial point of view, given a binary\nword~$b$, the map~$\\beta$ computes the sum of the binary trees having~$b$\nas canopy. The composition $\\rho \\circ \\beta$ is an injective Hopf map\nfrom~${\\bf Sym}$ to~${\\bf Baxter}$. From a combinatorial point of view, given a\nbinary word~$b$, the map $\\rho \\circ \\beta$ computes the sum of the pairs\nof twin binary trees~$(T_L, T_R)$ where the canopy of~$T_R$ is~$b$ and\nthe canopy of~$T_L$ is the complementary of~$b$.\n\n\\subsubsection{Full diagram of embeddings}\nFigure~\\ref{fig:DiagrammeAHC} summarizes the relations between known Hopf\nalgebras related to ${\\bf Baxter}$.\n\\begin{figure}[ht]\n    \\centering\n    \\begin{tikzpicture}[scale=.5]\n        \\node(FQSym)at(5,0){${\\bf FQSym}$};\n        \\node(DSym4)at(0,-3){$\\DSym{4}$};\n        \\node(Baxter)at(5,-3){${\\bf Baxter}$};\n        \\node(DSym3)at(0,-6){$\\DSym{3}$};\n        \\node(PBT)at(10,-6){${\\bf PBT}$};\n        \\node(Sym)at(5,-9){${\\bf Sym}$};\n        %\n        \\draw[Injection,dashed](DSym4)--(FQSym);\n        \\draw[Injection](Baxter) edge node[anchor=south,right] {$\\theta$} (FQSym);\n        \\draw[Injection](DSym3)--(DSym4);\n        \\draw[Injection](DSym3) edge node[anchor=south,below] {$\\alpha$} (Baxter);\n        \\draw[Injection](PBT) edge node[anchor=south,below] {$\\rho$} (Baxter);\n        \\draw[Injection](Sym)--(DSym3);\n        \\draw[Injection](Sym) edge node[anchor=south,below] {$\\beta$} (PBT);\n    \\end{tikzpicture}\n    \\caption{Diagram of injective Hopf maps between some Hopf algebras\n    related to ${\\bf Baxter}$. Arrows~$\\rightarrowtail$ are injective Hopf maps.}\n    \\label{fig:DiagrammeAHC}\n\\end{figure}\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Observations}\n\n\\noindent\nDust enshrouded activity of a galaxy can be studied ideally by\nmid--infrared (MIR) observations. To explore the origin of the nuclear\nMIR emission of galaxies as being due to either active galactic nuclei\n(AGN) or star formation, observations of high spatial resolution are\nrequired.\n\n\\noindent\nThe nuclear MIR surface brightness is introduced as a quantitative\nmeasurement for AGN and starburst activity. However, one is unable to\ndistinguish between both activity types using the nuclear MIR surface\nbrightness derived from 4m class telescopes, even when adopting the\ntheoretical diffraction limit of $0.7''$ (FWHM) of such telescopes\n(cmp. small gray symbols in Fig.~\\ref{surf.ps}).  Since the PSF width\nis twice as large as for a 8m class telescope and the point source\nsensitivity is a factor 16 lower, it becomes more difficult for a 4m\nto resolve starburst and the surface brightness of unresolved sources\nis reduced. Data recently obtained at 8m class telescopes\n(Siebenmorgen et al. 2008) show that, out to a distance of 100Mpc, the\nMIR surface brightness acquired clearly differentiate AGN from SB\nbehavior (Fig.~\\ref{surf.ps}). Utilizing VISIR at the VLT the AGN\nstill appear point like whereas most starburst are resolved in the\nMIR.  This discrimination was made possible by an increase in spatial\nresolution by a factor 2. Therefore it provides a clue to what will be\npossible by increasing the spatial resolution by another factor 5 when\ngoing from the VLT to the proposed extreme large telescope such as the\nE-ELT which will be 40m class. For the E-ELT a mid infrared instrument\nis included in the instrumentation plan and it has beside imaging also\nhigh resolution spectroscopic and polarimetric observing capabilities\n(Brandl et al., 2010).\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=11cm,angle=0]{extin.ps}\n\\caption{Mean extinction curve of the ISM by Fritzpatrick\\&Massa\n  (2007, black) and a fit (magenta) by the dust model of\n  Sect.~\\ref{dust}. Individual dust components are shown as\n  labeled. The grey areas indicate the 1$\\sigma$ deviation as of the\n  samples.\n\\label{dust.ps} }\n\\end{center}\n\\end{figure}\n\n\n\\section{Dust model \\label{dust}}\n\n\n\\noindent \nTeams interested in modelling the processing of radiation by dust in\ngalaxies often apply a dust model as derived for the diffuse ISM of\nthe Milky Way.  Dust cross sections are computed using similar optical\nconstants and temperature fluctuating particles such as PAHs are\nincluded.  We developed one of such dust models in which {\\it {large}}\n($60\\rm{\\AA}<a<0.2-0.3\\mu$m) silicate (Draine 2003) and amorphous\ncarbon (Zubko et al. 2004) grains and {\\it {small}} graphite\n($5\\rm{\\AA}<a<80$\\,\\AA) grains are considered.  We apply a power law\nsize distribution: $n(a) \\propto a^{-3.5}$ and absorption and\nscattering cross-sections are computed with Mie theory.  In addition\nthere are {\\it {PAHs}} with 30 and 200 C atoms with absorption cross\nsection as given by Schutte et al. (1993).  By computing cross\nsections above 100\\,eV, we consider an approximation of kinetic energy\nlosses (Dwek\\& Smith 1996) and apply it to all particles. The choice\nof parameters is set up to achieve a fit of the mean extinction curve\nof the ISM (Fitzpatrick \\& Massa 2007), as shown in Fig.\\ref{dust.ps}.\n\n\\noindent \nWith the advent of {\\it{ISO}} and {\\it{Spitzer}} more PAH emission\nfeatures and more details of their band structures are detected\n(Tielens 2008).  We consider 17 emission bands and take Lorentzian\nprofiles (Siebenmorgen et al. 1998). Parameters of what we call\nastronomical PAH are calibrated using mid-IR spectra of starburst\nnuclei and the RT model as of Sect.~{\\ref{sb}}. PAH cross sections of\nthe emission bands are listed by Siebenmorgen \\& Kr\\\"ugel (2001). In\nthe model, we use dust abundances of [X]\/[H] (ppm) of: 31 for [Si],\n150 [amorphous C], 50 [graphite] and 30 [PAH], respectively; which is\nin agreement with cosmic abundance constraints (Asplund et\nal. 2009). We are in the process of upgrading the model to be\nconsistent with the polarization of the ISM (Voshchinnikov\n2004). Besides extinction the model accounts for the diffuse emission\nof solar neighborhood when the dust is heated by the interstellar\nradiation field (Mathis et al. 1983).\n\n\\section{Starbursts \\label{sb}}\n\n\\noindent \nThere are three different ways in the literature trying to reproduce\nthe SED of extra-galactic nuclei.  A first one uses a SED of a\nwell-known galaxy as a template to match other objects (Lutz et\nal. 2002). A second group reproduce the shape of the SED by optical\nthin dust emission using a scaled up version of the interstellar\nradiation field (Draine \\& Lee 2007). A third and more ambitious\nmethod is to solve the radiative transfer (RT) problem using\nassumptions about the galaxy.  The latter is done at various levels of\nsophistication and it may be instructive to point out technical\ndifferences. Teams solving the RT problem evaluate the emission from a\ndusty medium of spheroidal shape filled with stars and dust.  At first\nglance, the model results appear to agree, but upon closer inspection\none finds that deviations of derived parameters are substantial.  For\nexample for Arp220 Grooves et al.  finds an optical depth of a few\nwhereas we derive values between 70 -- 120.  We admit that we did not\nalways find it easy to pin down exactly which approximations our\ncolleagues used, still we try to summarize the main features of some\nRT models which are in widespread use.  Monte Carlo techniques are\ndiscussed in Sect.~\\ref{agn}.  We use the term {\\it {dust\n    self--absorption}} when photons which are emitted by a dust\nparticles may be absorbed by other dust particles within the model\nsphere and we call {\\it {exact RT}} when the RT equation is solved\naccurately including multiple scattering and dust self--absorption.\n\n{$-$} Groves (2004, this volume): Shock, photoionization and dust\nradiative transfer code called MAPPINGIII. The dust is distributed in\na screen and the RT is solved in one dimension ignoring dust\nself-absorption and treating scattering in forward direction only.\n\n{$-$} Efstathiou \\& Rowan--Robinson (2003), Efstathiou (this volume):\nDust is distributed in spherical symmetry. There are two components:\nmolecular clouds with exact RT computation and cirrus which is added\nas a foreground screen. Both components are uncoupled in the RT\nequation. The code includes a free parameter to scope with the\nobserved optical and UV spectrum.\n\n{$-$} Takagi et al. (2003, this volume): Exact RT in spherical\nsymmetry where dust and stars are homogeneously distributed. Molecular\nclouds (clumps) are not treated. The code includes a free parameter to\nscope with the observed optical and UV spectrum.\n\n{$-$} Siebenmorgen et al. (2007): Exact RT in spherical symmetry where\ndust and a young and old population of stars are distributed in the\ngalaxy. A fraction of the young stars are in molecular clouds for\nwhich a second exact RT computation is solved. The coupling of both RT\nsolutions, that of the galaxy and the embedded sources, is treated and\nthis is a particular feature of the model.\n\n{$-$} Silva et al. (1998): Present a code called GRASIL in which dust\nis distributed in axial-symmetry.  There is a molecular cloud\ncomponent with exact RT. A cirrus component is added in which the RT\nis solved by ignoring dust self-absorption and in which dust\nscattering is simplified by altering the dust absorption cross\nsection. In addition the code includes a free parameter to scope with\nthe observed optical and UV spectrum. All three components are\nuncoupled in the RT equation.\n\n{$-$} Popescu et al. (2002, this volume): Dust is distributed in\naxial-symmetry.  The model is fine tuned to Spiral galaxies\n(Sect.~\\ref{spirals}) and includes a bulge, two galactic disk\ncomponents and molecular clouds (clumps).  The RT is solved by\nignoring dust self-absorption and only the first scattering event is\ntreated. Clumps are added by a pre-computed template spectrum.  There\nis a free parameter to scope with the observed optical and UV\nspectrum. All five components are uncoupled in the RT equation. \n\n\n\\noindent \nAs the coupling between the RT calculations of the galaxy and the\nembedded sources is a computer intensive problem we provide a library\nof some 7000 SEDs {\\footnote{SED library available at: {\\tt\n      {http:\/\/www.eso.org\/$\\sim$rsiebenm\/sb\\_models\/}}}} for the\nnuclei of starburst and ultra--luminous galaxies (Siebenmorgen et al.,\n2007).  Its purpose is to quickly obtain estimates of the basic\nparameters of the object, such as luminosity, size and dust or gas\nmass and to predict the flux at yet unobserved wavelengths using a\nphysical model. Unfortunately, for faint or red-shifted objects\nphotometry is sometimes only provided at two MIR bands, for example at\n8 and 24$\\mu$m from Spitzer. In Fig.~\\ref{2dat.ps} we demonstrate for\nthe ULIRG NGC6240 that the SED library can be used to estimate the\ntotal luminosity to within a factor $\\sim 2$ in case only two such MIR\nfluxes are known.  We also see (Fig.~\\ref{2dat.ps}) that the SED will\nbe quite well constrained by an additional submm data point.  The\nmodel is applied to high red shifts ($z \\approx 3$, Efstathiou \\&\nSiebenmorgen 2009) where PAH have been detected.\n\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8cm,angle=90]{2datN6240.ps}\n\\caption{Elements of the SED starburst library by Siebenmorgen et\n  al. (2007, dotted) which fit photometry (circles) of NGC6240 at\n  8$\\mu$m (Siebenmorgen et al. 2004) and 24$\\mu$m (Klaas et al. 2001)\n  to within 30\\%; submm data by Benford (1999) and Klaas et\n  al. (2001), best fit is indicated as thick line \\label{2dat.ps}. }\n\\end{center}\n\\end{figure}\n\n\n\\section{Spirals \\label{spirals}}\n\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=10.5cm,angle=270]{scheSpiral.ps}\n\\end{center}\n\\vspace*{-1.0 cm}\n\\caption{Schematic view of the geometry applied in the models\nof Spirals by Popescu et al. \\label{scheSpiral.ps} }\n\n\\end{figure}\n\n\\noindent\nThe geometrical distribution of dust in spiral galaxies has been\ninvestigated by fitting images in the optical\/NIR.  This is done by\nmeans of 2d radiative transfer codes which include a bulge component\nand an old stellar population of stars; originally only dust\nabsorption and scattering was treated (Kylafis \\& Bahcall 1987,\nXilouris et al. 1999, Misiriotis et al., 2000). Building into the\nmodels typical dust emission properties gives a puzzle when compared\nto observations in the FIR. Typically the models underestimate the FIR\nluminosities by a factor of 3.  The FIR excess could be explained by\nPopescu et al. (2000, 2010, this volume) introducing more components.\nIn their models there is a distribution of diffuse dust associated\nwith the old and young stellar disk populations as well as a clumpy\ncomponent arising from dust in the parent molecular clouds in star\nforming regions. Basic parameters of their models are: angular size\nand inclination of the disk, the central face-on dust opacity in the\nB-band, a clumpiness factor for the star-forming regions, the\nstar-formation rate, the normalized luminosity of the stellar\npopulation and the bulge-to-disk ratio and a wavelength dependent\nescape probability of stellar radiation (which is sometimes treated as\nmodel output). A schematic view of the geometry of the RT model is\npresented in Fig.~\\ref{scheSpiral.ps}. The model is successfully\napplied to a large sample of galaxies from the Millennium Galaxy\nCatalog Survey (Driver et al. 2007). The observed\nattenuation--inclination relation is fit when a two dust disks are\nconsidered whereas data are not fit in a single disk scenario.\n\nOver the past decade the edge on spiral NGC891 become a benchmark test\nfor RT models of spiral galaxies. Bianchi (2008) uses a clumpy disks\nmodel and applied MC techniques which are of advantage when dealing\nwith such complicated geometries (see Sect.~\\ref{agn}). His results\ngive support to the idea that the diffuse dust disk is more extended\nthan the stellar disks. De Looze et al. (20011, this volume) present a\nMC radiative transfer model of the Sombrero galaxy which is able to\nfit the SED and optical\/NIR images and extinction profiles. Using only\nan old stellar population the dust luminosity in the FIR is again\nunderestimated by a factor 3.  The discrepancy is solved by assuming a\nstar forming stellar component both in the ring and inner disk to\naccount for emission at 24 and 70$\\mu$m. In the submm an additional\ndust component is used, accounting for 75\\% of the total dust content\nand distributed in quiescent compact clumps. A possibility that part\nof the dust could be composed of grains with a higher submm\nemissivity, of for example large fluffy grains (Kr\\\"ugel \\&\nSiebenmorgen 1994), could not be ruled out. On the other hand a\nsimilar approach introducing a clumpy medium in a single disk model is\napplied to the edge-on spiral galaxy UGC4754 (Baes et al. 2011). The\nmodel has a deficit in the FIR luminosity and the authors propose\nhigher submm dust emissivities to solve the energy ballance at such\nlong wavelengths.\n\n\n\n\\section{AGN and Monte Carlo \\label{agn}}\n\n\\noindent\nDust is detected in the majority of active galactic nuclei (Haas et\nal., 2008).  According to the unified scheme (Antonucci \\& Miller,\n1985), AGN are surrounded by a dust obscuring torus. However,\nobservations are not able to resolve the inner parts of AGNs so that\nthe geometrical distribution of the dust is a matter of\ndebate. Theoretical considerations favor a clumpy structure in a torus\nlike configuration of optical thick dust clouds surrounding the black\nhole accretion disk (Pier \\& Krolik (1992). Some evidence of a clumpy\nor filamentary structure of the torus is given by VLTI observations of\nthe nearby active galactic nuclei in Circinus (Tristram et al. 2007).\nRadiative transfer models of homogeneous toroidal structures\nover-predict the silicate absorption and emission band at around\n10$\\mu$m when compared to observations. The silicate emission feature\nin type I AGN is rather shallow (Siebenmorgen et al. 2005). A\nstatistical attempt to describe the radiation from a clumpy AGN torus\nis given by Nenkova et al. (2002) and more detailed radiative transfer\ncomputations using the Monte Carlo technique are presented by H\\\"onig\net al. (2006) and Schartmann et al. (2008).\n\nWe develop a vectorized three dimensional Monte Carlo (MC) technique\nto solve the radiative transfer problem in such a fairly complicated\ngeometry of a clumpy dust torus. Originally the MC radiative transfer\nmethod for scattered light in astronomical sources is presented by\nWitt (1977) and important improvements are introduced by Lucy\n(1999). A first version of the code which we are using is developed by\nKr\\\"ugel (2006). To reduce the computational effort of MC dust\nradiative transfer methods different optimization strategies are\ndeveloped (Lucy 1999, Bjorkman \\& Wood 2001, Gordon et al. 2001,\nMisselt et al. 2001, Baes 2008, Bianchi 2008, Baes et al. 2011). A\nnumerical solution of the radiative transfer equation which is\nspecifically developed to be vectorized and to run on graphical\ncomputer units (GPU) is presented by Heymann (2010). A GPU version\nincluding stochastic heated grains is presented (Siebenmorgen et\nal. 2010), in which some detailed physics of the destruction of PAHs\nby soft (photospheric) and hard (X--ray) radiation components is\ntreated, (see Siebenmorgen \\& Kr\\\"ugel 2010 for a discussion of PAH\ndissociation).  The speed up factor of the GPU method is proportional\nto the number of processors available. Therefore the GPU scheme is on\nour standard PC with a conventional graphic card about 100 times\nfaster when compared to the original scalar version of the MC program.\nThe numerical solution provides the temperature of the dust, spectra\nand images within acceptable timescales. The code is tested against\nexisting benchmark results. The method handles arbitrary dust\ndistributions in a three dimensional Cartesian model space at various\noptical depths. Therefore it is well adopted to be applied to a clumpy\ntorus structure around an AGN.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=6cm,angle=0]{agnTklump.eps}\n\\end{center}\n\\caption{Zoom into the clumpy AGN torus displaying the temperature\n  structure of individual clouds. \\label{agnTklump} }\n\\end{figure}\n\n\n\\begin{figure}[htb]\n\\begin{center}\n \\includegraphics[width=13.5cm,angle=0]{torus_emm_sca.eps}\n\\caption{Intensity maps of the clumpy AGN dust torus model\n  (color-code: cgs units normalized to the peak intensity).  The model\n  is made up of 1000 clumps and each clump has a total optical depth\n  of $\\tau_{\\rm V} = 30$.  Mid IR images at 10\\,$\\mu$m are dominated\n  by dust emission and displayed in the top panel. The optical light\n  at 0.55\\,$\\mu$m represents the scattering intensities and are shown\n  in the bottom panel.  The AGN torus is viewed face-on ($\\theta =\n  0^\\circ$, left), at $\\theta = 45^\\circ$ (middle) and edge-on\n  ($\\theta = 90^\\circ$, right).  \\label{agn.ps} } \n\\end{center}\n\\end{figure}\n\n\n\n\nThe primary heating source of the dust in the torus emerges from the\naccretion disc around the massive black hole. For the spectral shape\nwe apply a broken power law as suggested by Rowan-Robinson (1995). As\nexample we use a total AGN luminosity of $L_{\\rm {AGN}} =\n10^{45}$\\,erg\/s.  This sets the inner radius $R_{\\rm {in}} \\sim 0.4\n\\sqrt{L_{\\rm {AGN}}\/10^{45}}$\\,[pc] where the dust evaporation zone is\nlocated. We treat the torus to an outer radius of $R_{\\rm {out}} = 50\nR_{\\rm {in}}$ and consider a torus opening angle of $\u03b8 \\sim 20^{\\rm\n  o}$.  Clouds are randomly distributed in the otherwise optical thin\ndust torus. Individual clouds are approximated with a sphere of\nconstant density and radius of 3\\,pc. The optical depth through the\ncenter of the clouds is $\\tau_{\\rm V} = 30$. In Fig.~{\\ref{agnTklump}}\nwe zoom into a part of the clumpy torus and show the temperature\nstructure of the clouds. The individual hemispheres which are oriented\nin direction of the central heating source are warmer as compared to\ntheir dark sides. Shadowed clumps become also colder than those\ndirectly heated by the central engine. Intensity maps of the emission\nat 10$\\mu$m and scattered light at 0.5$\\mu$m of the clumpy AGN dust\ntorus model are displayed in Fig.~\\ref{agn.ps}. The maps show that in\na clumpy medium even in the edge on case some radiation from the\ncentral region is visible.  A more detailed description of the clumpy\nAGN torus model and a further application to the silicate emission in\nQuasars is given in Heymann \\& Siebenmorgen (2011).\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA strange attractor   is an invariant set with at least one positive Lyapunov exponent whose basin of attraction has non-empty interior. Nowadays, at least for families of dissipative systems, chaotic dynamics is mostly understood as the persistence of strange attractors\n--- occurring for parameters in a set of  positive Lebesgue measure.\n Persistence of dynamics is physically relevant because it means that the phenomenon is \\emph{observable} with positive probability. \nProof of the existence of strange attractors is usually obtained by comparing the dynamics in an invariant set to either the Lorenz or the H\\'enon attractors, or through the unfolding of a singularity,\n for a discussion see \\cite{BIRR}.\n\n\nThere are few examples of periodically-forced vector fields exhibiting complex dynamics that may be proven analytically. In this article, we give an explicit mechanism to obtain strange attractors in a two-parameter family of vector fields unfolding an attracting heteroclinic network. When the first parameter is different from zero, two normally hyperbolic attracting tori arise near the network. Fixing this parameter and varying the second, the tori break and suspended horseshoes emerge, via the \\emph{torus-breakdown} phenomenon \\cite{AS91, Anishchenko}. In the meantime, persistent attractors (of H\\'enon-type) associated to homoclinic tangencies are created. \n\n \nOur  \\emph{route to chaos}   from an attracting network is different from the routes described by  \\cite{Kaneko} and \\cite{Bakri}, where the authors use coupled oscillators. Another different itinerary has been described by \\cite{FJ2020} in the context of the \\emph{Langford system}. These works are discussed in Section \\ref{discussion}. \n\n\nThe theory developed in this paper is explicitly applicable to the analysis of various specific differential equations and the results obtained are beyond the capacity of the classical Birkhoff-Melnikov-Smale method associated to \\emph{heteroclinic tangles} \\cite{Wang_2013}.\n\nOur purpose in writing this paper is not only to point out the range of phenomena that can occur when simple non-linear equations are periodically forced, but to bring to the foreground the techniques that have allowed us to reach these conclusions in a relatively straightforward manner. \n\n\n\nWe analyse a family of periodic perturbations of an attracting symmetric heteroclinic network defined on the two-sphere. Instead of looking at the time-$T$ maps as in \\cite{LR18, TD1}, we extend the phase space and we explicitly compute return maps induced by the perturbed equations in\na neighborhood of the extended heteroclinic network. The  cyclic variable's speed plays an important role in the emergence of the horseshoes.\n Using the techniques explored in  \\cite{Shilnikov_book_1, TS86}, we reduce the analysis of the non-autonomous system to that of a two-dimensional map on a circloid.  These techniques are clearly not limited to the systems considered here. It is our hope that they will find applications in other dynamical systems, particularly those that arise naturally from mechanics or physics \\cite{AHL2001, DT3, Ruelle, TD1, Rabinovich06}. See also the dynamical description of the periodically-forced van der Pol oscillator in   \\cite{Shilnikov_book_1, TS86}, where the authors used the \\emph{Afraimovich Annulus Principle} to prove the existence of an invariant torus. \n \nThe title \\emph{``Periodic forcing of a heteroclinic network''} refers to the study of the dynamics associated to a parametric periodically-forced vector field unfolding  an asymptotically stable heteroclinic network.\n\n\n\\subsection*{Structure of the article}\nThis article is organized as follows. In Section \\ref{sec-setting}, we describe the setting of our problem\nand state the main results after having revised some conceptual preliminaries  in Section \\ref{preliminaries}. \nResults on general families of vector fields are proved in Section~\\ref{secProva_G}, after the derivation  in Section \\ref{secPrelim} of the first return map to a given cross section.\n An explicit example is treated in Section~\\ref{PropA}.\nWe finish this article with a discussion on the way our results fit in literature in Section \\ref{discussion}.\n\n\\section{Preliminaries}\n\\label{preliminaries}\nIn this section, we introduce some terminology for vector fields acting on three-dimensional Riemannian manifolds that we will use in the remaining sections.\nConsider the two-parameter family of $C^3$--smooth autonomous differential equations\n\\begin{equation}\n\\label{general2a}\n\\dot{x}=F_{(\\nu, \\mu)}(x)\\qquad x\\in {\\mathbf S}^3\\subset {\\mathbf R}^4  \\qquad \\nu, \\mu\\in{\\mathbf R}\n\\end{equation}\nwhere ${\\mathbf S}^3$ denotes the unit sphere, endowed with the usual topology. Denote by $\\varphi_{(\\nu, \\mu)}(t,x)$, $t \\in {\\mathbf R}$, the associated flow. The flow is \\emph{complete} (\\emph{i.e.}  solutions are defined for all $t\\in {\\mathbf R}$) because  ${\\mathbf S}^3$ is a compact manifold without boundary.\n \n \\subsection{Heteroclinic structures and symmetry}\nSuppose that $A$ and $B$ are two hyperbolic saddles of \\eqref{general2a}. There is a {\\em heteroclinic cycle} associated to $A$ and $B$ if\n$$W^{u}(A)\\cap W^{s}(B)\\neq \\emptyset \\qquad \\text{and} \\qquad W^{u}(B)\\cap W^{s}(A)\\neq \\emptyset.$$ \n\nThe non-empty intersection of $W^{u}(A)$ with $W^{s}(B)$ is called a \\emph{heteroclinic connection} between $A$ and $B$, and will be denoted by $[A \\rightarrow  B]$. Although the existence of heteroclinic cycles may be a non-generic feature within differential equations, they may be structurally stable within families of systems which are equivariant under the action of a compact Lie group $\\mathcal{G}\\subset \\mathbb{O}(4)$, due to the existence of flow-invariant subspaces (see  \\cite{GH}).\n\n \nGiven a group $\\mathcal{G}$ of endomorphisms of ${\\mathbf S}^3\\subset {\\mathbf R}^4$, we will consider two-parameter families of vector fields $\\left(F_{(\\nu, \\mu)}\\right)$ under the equivariance assumption $$F_{(\\nu, \\mu )}(\\gamma x)=\\gamma F_{(\\nu, \\mu )}(x)$$ for all $x \\in {\\mathbf S}^3$, $\\gamma \\in \\mathcal{G}$ and $(\\nu, \\mu )\\in  {\\mathbf R}^2.$\nFor an isotropy subgroup $\\widetilde{\\mathcal{G}}< \\mathcal{G}$, we will write $\\Fix(\\widetilde{\\mathcal{G}})$ for the vector subspace of points that are fixed by the elements of $\\widetilde{\\mathcal{G}}$. For $\\mathcal{G}-$equivariant differential equations, the subspace $\\Fix(\\widetilde{\\mathcal{G}})$ is flow-invariant.\n\nIf $\\Omega\\subset {\\mathbf S}^3$ is a flow-invariant set of \\eqref{general2a}, its basin of attraction, $\\mathcal{B}(\\Omega)$, is the set of points in ${\\mathbf S}^3$ whose orbits have $\\omega-$limit in $\\Omega$. We say that $\\Omega$ is \\emph{asymptotically stable} if $\\mathcal{B}(\\Omega) $ contains \n all the half-trajectories in positive time starting in\nan open neighbourhood of $\\Omega$. \n \\subsection{Rotational horseshoes}\nLet $\\mathcal{H} $ stand for the infinite annulus $\\mathcal{H} = {\\mathbf S}^1 \\times {\\mathbf R}$ endowed with the usual inner product from ${\\mathbf R}^2$. We denote by $Homeo^+(\\mathcal{H} )$ the set of homeomorphisms of the annulus which preserve orientation.\nGiven a homeomorphism $f :X \\rightarrow X$  and a partition of $m\\geq 2$ elements $R_1,..., R_{m}$ of $X\\subset \\mathcal{H}$, the itinerary  function $\\xi: X \\rightarrow \\{1, ..., m\\}^{\\mathbf Z}= \\Sigma_m$ is defined by $$\\xi(x)(j)=k\\quad   \\Leftrightarrow \\quad f^j(x)\\in R_k, \\quad \\text{for every} \\quad j\\in {\\mathbf Z}.$$  \nFollowing \\cite{Passeggi}, we say that a compact invariant set $\\Lambda \\subset \\mathcal{H} $ of $f \\in Homeo^+(\\mathcal{H} )$ is a \\emph{rotational horseshoe} if it admits a finite partition $P =\\{R_1, ..., R_{m} \\}$  by sets  $R_i$ with non empty interior in $\\Lambda$ so that\n\\begin{itemize}\n\\item \nthe itinerary $\\xi$ defines a semi-conjugacy between $f|_\\Lambda$ and the full-shift $\\sigma: \\Sigma_m \\rightarrow \\Sigma_m$, that is $\\xi  \\circ f = \\sigma \\circ \\xi$ with $\\xi$ continuous and onto;\n\\item \nfor any lift $G: {\\mathbf R}^2 \\rightarrow {\\mathbf R}^2$ of $f$, there exist  $k>0$ and $m$ vectors $v_1, ...,v_{m} \\in {\\mathbf Z} \\times \\{0\\}$ so that\n$$\n\\left\\| (G^n(\\hat{x})-\\hat{x})  - \\sum_{i=0}^n v_{\\xi(x)(i)}\\right\\| <k \\qquad \\text{for every} \\qquad  \\hat{x}\\in \\pi^{-1}(\\Lambda), \\quad n\\in {\\mathbf N},\n$$\nwhere $\\|\\star\\|$ denotes the usual norm of ${\\mathbf R}^2$,  $\\pi:{\\mathbf R}^2\\rightarrow \\mathcal{H}$ denotes the usual projection map and $\\hat{x} \\in \\pi^{-1}(\\Lambda)$ is the lift of $x$; more details in the proof of Lemma 3.1 of \\cite{Passeggi}. The existence of a rotational horseshoe for a map implies positive topological entropy  at least  $ \\log m $.\n\\end{itemize}\n\n\\subsection{Strange attractors}\n\nStrange attractors contribute to the richness and complexity of a dynamical system.  We introduce the following notion, adapted from \\cite{MV93}, to the situation under consideration.\nA  (H\\'enon-type) \\emph{strange attractor} of a two-dimensional dissipative diffeomorphism $f$ defined in a Riemannian manifold, is a compact invariant set $\\Lambda$ with the following properties:\n\\begin{itemize}\n\\item \n$\\Lambda$  equals the closure of the unstable manifold of a hyperbolic periodic point;\n\\item \nthe basin of attraction of $\\Lambda$   contains an open set (and thus has positive Lebesgue measure);\n\\item \nthere is a dense orbit in $\\Lambda$ with a positive Lyapounov exponent (exponential growth of the derivative along the orbit).\n\\end{itemize}\nA vector field possesses a strange attractor if the first return map to a cross section does.  \n \n\n\\section{Setting and main results}\n\\label{sec-setting}\nOur object of study is the dynamics around an attracting  heteroclinic network for which we give a rigorous description here.  \\subsection{The object of study}\nConsider the  family of $C^3$--autonomous differential equations on a two dimensional manifold ${\\mathcal M}^2$ diffeomorphic to the two-sphere ${\\mathbf S}^2\\subset {\\mathbf R}^3$ and parametrised by $\\nu\\in{\\mathbf R}$\n\\begin{equation}\n\\label{general0}\n \\begin{array}{ll}\n\\dot x={\\mathcal F}_\\nu(x), \\quad x\\in {\\mathbf R}^3.\n\\end{array} \n \\end{equation}\n\n\nSuppose that, for $\\nu=0$, the flow of \\eqref{general0} has an attracting heteroclinic cycle associated to two\nequilibria\nand that for $\\nu\\ne 0$ one of the heteroclinic connections is broken, yielding an attracting periodic solution, as illustrated in Figure \\ref{scheme1}.\nMore precisely we are assuming that \\eqref{general0}   satisfies: \n \n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\textbf{(A\\arabic{enumi})}}\n\\renewcommand{\\labelenumi}{\\textbf{\\theenumi}}\n\\item\\label{A1}\n  there is  a flow-invariant manifold ${\\mathcal M}^2$ diffeomorphic to ${\\mathbf S}^2$ that is attracting in the sense that every nearby trajectory is asymptotic to it in forward time; \n\\item \\label{A2}\nthere are  two equilibria of saddle type\n${\\mathbf v}$ and ${\\mathbf w}$;\n\\item \\label{A3}\nthere are  two heteroclinic connections from ${\\mathbf w}$ to ${\\mathbf v}$, forming a Jordan curve $\\vartheta$ on ${\\mathcal M}^2$.\n\\setcounter{lixo}{\\value{enumi}}\n\\end{enumerate}\nMoreover, the restriction of the flow of \\eqref{general0}  to ${\\mathcal M}^2$ satisfies: \n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\textbf{(A\\arabic{enumi})}}\n\\renewcommand{\\labelenumi}{{\\theenumi}}\n \\setcounter{enumi}{\\value{lixo}}\n\\item \\label{A4}\nfor $\\nu=0$ there are two heteroclinic connections from ${\\mathbf v}$ to ${\\mathbf w}$;\n\\item \\label{A5}\nfor $\\nu=0$ the heteroclinic network formed by the connections between ${\\mathbf v}$ and ${\\mathbf w}$ is attracting;\n\\item \\label{A6}\nfor $\\nu=0$ the only periodic solutions are the equilibria; \n\\item\\label{A7} for $\\nu\\ne 0$ the heteroclinic connections from ${\\mathbf v}$ to ${\\mathbf w}$ are broken and there are two attracting hyperbolic periodic solutions, each one in one connected component of   ${\\mathcal M}^2\\backslash \\vartheta$. \n\\setcounter{lixo}{\\value{enumi}}\n\\end{enumerate}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=3.5cm]{scheme1_LR.pdf}\n\\end{center}\n\\caption{\\small \nPartial phase portrait associated to  the one-parameter family \\eqref{general0},\na periodic solution appears when a heteroclinic connection is broken. (A): $\\nu=0$; (B): $\\nu\\neq 0$.  }\n \\label{scheme1}\n\\end{figure}\n \nThe period of the  solutions of  \\ref{A7}  tends to infinity when $\\nu$ tends to zero, as they accumulate on the heteroclinic cycles containing ${\\mathbf v}$ and ${\\mathbf w}$. From \\ref{A6}, these are the only non-constant periodic solutions. Property~\\ref{A7} is the generic unfolding of a heteroclinic network like the one in \\ref{A5}. \nIn contrast, Property ~\\ref{A3} is not generic but occurs generically in systems with a flow-invariant two-dimensional manifold, for instance in the presence of symmetry --- see the example in \\S \\ref{sec_example} below.\n\n \nWe subject the autonomous differential equation \\eqref{general0}  to a family of non-constant  time-periodic perturbations $\\phi_\\mu(t,x)$ of period $\\pi\/\\omega\\in{\\mathbf R}^+$, where the parameter $\\mu\\ge 0$  controls the amplitude of the perturbation. \nThe perturbed equation \n$$\n\\dot x={\\mathcal F}_\\nu(x)+\\phi_\\mu(t,x),\\qquad  x\\in  {\\mathbf S}^2, \nt\\in {\\mathbf R}\n$$ \nmay be converted into an autonomous equation \n$$\n(\\dot x,\\dot\\theta)= F_{(\\nu,\\mu)} (x,\\theta)\n$$ \nin ${\\mathcal M}^2\\times {\\mathbf S}^1$ by rewriting it as\n\\begin{equation}\n\\label{general}\n\\left\\{\\begin{array}{ll}\n\\dot x={\\mathcal F}_\\nu(x)+\\phi_\\mu(\\theta,x)&{}\\\\\n\\dot\\theta={2\\omega}&\\pmod{2\\pi}.\n\\end{array}\\right.\n\\qquad\\qquad \\phi_0(\\theta,x)\\equiv 0. \n\\end{equation}\n For $\\mu=0$ the last equation of \\eqref{general} is not coupled to the first. Hence, the dynamics of \n\\eqref{general} may be obtained from conditions \\ref{A1}--\\ref{A7}, as follows.\n\n\\begin{Lprop}\n\\label{PropB1}\nIf \\ref{A1}--\\ref{A7} hold for  ${\\mathcal F}_\\nu$\nthen the flow of \\eqref{general} for $\\mu=0$ and  small $\\nu\\ge 0$ satisfies:\n\\begin{enumerate}\n \\item  there is  an invariant flow-invariant manifold ${\\mathcal M}^3$  diffeomorphic to ${\\mathbf S}^2\\times {\\mathbf S}^1$\nthat is globally attracting, in the sense that it attracts all trajectories in its neighbourhood.\n\\setcounter{maislixo}{\\value{enumi}}\n\\end{enumerate}\nFurthermore, the restriction of the flow to ${\\mathcal M}^3$ satisfies:\n\\begin{enumerate}\n \\setcounter{enumi}{\\value{maislixo}}\n \\item\nthere are two periodic solutions in ${\\mathcal P}_{\\mathbf v}=\\{{\\mathbf v}\\}\\times{\\mathbf S}^1$ and  ${\\mathcal P}_{\\mathbf w}=\\{{\\mathbf w}\\}\\times{\\mathbf S}^1$ of saddle type;\n\\item\nfor  $\\nu=0$ the invariant manifolds of ${\\mathcal P}_{\\mathbf v}$ and ${\\mathcal P}_{\\mathbf w}$ in ${\\mathcal M}^3$ coincide, forming  a  heteroclinic network $\\Gamma$ with the geometry of  a  singular two-dimensional torus of genus 2;\n\\item \n  for $\\nu>0$  the manifolds $W^u({\\mathcal P}_{\\mathbf w})$ and $W^s({\\mathcal P}_{\\mathbf v})$  in ${\\mathcal M}^3$ coincide;\n\\item \\label{L5}\n  for $\\nu>0$ we have  $W^u({\\mathcal P}_{\\mathbf v})\\cap W^s({\\mathcal P}_{\\mathbf w})=\\varnothing$;\n   \\item\\label{L6}\n  for $\\nu>0$  there are two attracting  invariant two-dimensional tori $\\mathcal{T}^\\pm(\\nu)$;\n \\item\\label{L7}\n when  $\\nu \\rightarrow 0$, the tori $\\mathcal{T}^\\pm(\\nu)$ accumulate on $\\Gamma$.\n\\end{enumerate}\n\\end{Lprop}\n\nThe proof of Proposition \\ref{PropB1} follows \nby combining the dynamics of \\eqref{general0} with the existence of a cyclic variable (see Chapter 4 of \\cite{Shilnikov_book_1}).\n\n\\subsection{The periodically forced system}\n\\label{munot0}\n\n\nFor $\\nu=\\mu=0$, let $\\Sigma$ be a cross section of the heteroclinic cycle $\\Gamma$ in ${\\mathcal M}^3$. \nThen $\\Sigma$ is also a cross section of \\eqref{general} for small $\\nu,\\mu\\ge 0$.\nLet ${\\mathcal R}_{(\\nu,\\mu)}$ be the first return map to $\\Sigma$, with respect to the flow defined by ${F}_{(\\nu, \\mu)}$.  Define also\n$$\n\\Omega_{(\\nu,\\mu)}=\\left\\{X \\in \\Sigma: {\\mathcal R}^n_{(\\nu,\\mu)} (X)\\in \\Sigma, \\quad \\forall n \\in {\\mathbf N} \\right\\}\\qquad \\text{and} \\qquad \\Lambda_{(\\nu,\\mu)}= \\bigcap_{n \\in {\\mathbf N}} {\\mathcal R}^n_{(\\nu,\\mu)} \\left(\\Omega_{(\\nu,\\mu)}\\right).\n$$\n In this article, we \n present a comprehensive analysis on the dynamics \nof ${\\mathcal R}_{(\\nu, \\mu)}$ on the \\emph{non-wandering set}\n $\\Lambda_{(\\nu, \\mu)}$. When there is no risk of misunderstanding, we omit the subscripts ${(\\nu,\\mu)}$.\n \nWhen $(\\nu,\\mu)\\ne(0,0)$  one expects that, generically, $W^u({\\mathcal P}_{\\mathbf v}) \\pitchfork W^s({\\mathcal P}_{\\mathbf w})$.\nThe case when $W^u({\\mathcal P}_{\\mathbf v})\\cap W^s({\\mathcal P}_{\\mathbf w})\\ne\\varnothing$  has been discussed in \\cite{LR17, Wang_2013}, here we are mostly concerned with the case $W^u({\\mathcal P}_{\\mathbf v}) \\cap W^s({\\mathcal P}_{\\mathbf w})=\\varnothing$. \nWe also suppose property \\ref{A3} (extended to equation \\eqref{general}) still holds for the forced system, hence our assumptions are: \n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\textbf{(A\\arabic{enumi})}}\n\\renewcommand{\\labelenumi}{{\\theenumi}}\n \\setcounter{enumi}{\\value{lixo}}\n\\item \\label{A8}\n$W^u({\\mathcal P}_{\\mathbf w}) = W^s({\\mathcal P}_{\\mathbf v})$; \n\\item \\label{A9}\n$W^u({\\mathcal P}_{\\mathbf v}) \\cap W^s({\\mathcal P}_{\\mathbf w})=\\varnothing$. \n\\end{enumerate}\n\n\nOur first main result is about the existence  of an invariant   set whose dynamics is conjugate to a full shift over a finite number of symbols. In addition, we also prove the existence of observable chaos.\n\n \n \\begin{Lth} \n  \\label{role_omega}\n  If \\ref{A1}--\\ref{A9} hold for \\eqref{general} then\n  \\begin{enumerate}\n  \\item for every small $\\nu,\\mu>0$ there exists $\\omega_0>0$ such that if $\\omega>\\omega_0$, then  $\\Lambda_{(\\nu, \\mu)}$ contains  an invariant set whose dynamics is conjugate to a full shift in two symbols;\n  \\item \nfor $(\\nu, \\mu)$ in a set  $\\mathcal{U}\\subset {\\mathbf R}^2$  of positive Lebesgue measure, the return map\n${\\mathcal R}_{(\\nu,\\mu)}$ exhibits a strange attractor.\n\\end{enumerate}\n \\end{Lth}\n \n   The dynamics of  $\\Lambda_{(\\nu, \\mu)}$ is mainly governed by the geometric configuration of the global invariant manifold $W^u({\\mathcal P}_{\\mathbf v})$.\n  The proof of  this result is  \n  done is Section~\\ref{secProva_G}.  Horseshoes of Theorem  \\ref{role_omega} have a different nature from those associated to the \\emph{heteroclinic tangle} in which the manifolds have a transverse intersection  \\cite{LR17, Wang_2013}.  This will be discussed in Section \\ref{discussion}.\n\nFor a fixed $\\nu>0$, if the ratio of $\\omega$ and the period of the hyperbolic periodic solution of  \\ref{A7}  is irrational, then trajectories on the torus $\\mathcal{T}^\\pm(\\nu)$ are unlocked, in the sense that they never close. These solution on  the torus are called \\emph{quasiperiodic} \\cite{Herman, Shilnikov_book_1}. \n  If the frequencies have a rational ratio,  trajectories are locked. \n  \n\nIn a resonant torus, where all solutions are locked, the frequency locking ratio $p\/q$ means that while the $x$ component of  a solution turns $p$ its $\\theta$ component winds $q$ times.\n This ratio is related  to the \\emph{rotation number} associated to the periodic orbit \\cite{Herman} and will be used in the proof of the second part of Theorem \\ref{role_omega}.\n\n\\subsection{The example}\n\\label{sec_example}\n An explicit two-parameter family ${\\mathcal F}_\\nu(x)+\\phi_\\mu(t,x)$ of vector fields   in ${\\mathbf S}^2\\subset {\\mathbf R}^3$  such that ${\\mathcal F}_\\nu(x)$ satisfies  \\ref{A1}--\\ref{A7}  is given by\n \\begin{equation}\n\\label{general4}\n\\left\\{ \n\\begin{array}{l}\n\\dot x_1 = x_1(1-r^2)-\\alpha x_1 x_3 +\\beta x_1x_3^2 + (1- x_1) \\,\\, \\,  (\\mu\\,  [f (\\theta)-1]+\\nu)\\\\\n\\dot x_2 =  x_2(1-r^2) + \\alpha x_2 x_3 + \\beta x_2 x_3^2 \\\\\n\\dot x_3 = x_3(1-r^2)-\\alpha(x_2^2-x_1^2)-\\beta x_3 (x_1^2+x_2^2) \\\\\n \\dot\\theta={2\\omega}\\pmod{2\\pi}\n\\end{array}\n\\right.\n\\end{equation}\nwhere \n$$\n\\nu, \\omega\\in {\\mathbf R}^+\\quad \\mu\\in{\\mathbf R}\\qquad\nr^2= x_1^2+x_2^2+x_3^2, \\qquad \\beta<0<\\alpha, \\qquad\n  |\\beta|<\\alpha ,\n$$\n and $f$ is a  non constant $2\\pi$-periodic map of class $C^3$.\n\n For $\\mu=\\nu=0$, the equation $\\dot x={\\mathcal F}_{0}(x)$, $x\\in {\\mathbf R}^3$,  is one of the examples constructed and analysed in   \\cite{ACL06} and also  studied in \\cite{LR18}.  \n The perturbing  term $(1-x_1) \\  [\\mu (f(2\\omega t) -1)+\\nu] $ appears only in the first coordinate\n  for two reasons. First, it simplifies the computations. Secondly, it allows comparison with previous work by other authors \\cite{AHL2001,DT3, Rabinovich06, TD1}. \n\n  \\begin{Lprop}\n\\label{periodic_solution_prop}\n The vector field ${\\mathcal F}_\\nu$ associated to  \\eqref{general4} at $\\mu=0$ is equivariant under the action of $\\kappa(x,y,z)=(x, -y,z)$ \nand therefore the plane $\\Fix({\\mathbf Z}_2(\\kappa))=(x,0,z)$ is flow-invariant.\nIf   $|\\nu|>0$ is small,  the flow of $\\dot\\zeta={\\mathcal F}_\\nu(\\zeta)$ satisfies conditions \\ref{A1}--\\ref{A7}.\nIn particular, the flow-invariant curve ${\\mathcal M}^2\\cap \\Fix({\\mathbf Z}_2(\\kappa))$ consists of  two equilibria of saddle-type ${\\mathbf v}$ and ${\\mathbf w}$ and two heteroclinic connections from ${\\mathbf w}$ to ${\\mathbf v}$. \nThere are also four equilibria in ${\\mathcal M}^2$  that are repelling foci and these are all the equilibria in ${\\mathcal M}^2$.\n\\end{Lprop}\n\n\n The proof of  this result is the content of Section~\\ref{PropA}.  \n \n\n From \\eqref{L6} in Proposition~\\ref{PropB1} it follows that there are two invariant tori for the flow of the equation $(\\dot x,\\dot\\theta)= F_{(\\nu,\\mu)} (x,\\theta)$\n associated to  \\eqref{general4}.\n Existence of invariant tori  is usually shown using the Afraimovich Annulus Principle \\cite{AS91}, here we show it directly by reducing the problem to a two-dimensional manifold and applying the Poincar\\'e-Bendixson theorem.\n \n From Theorem~\\ref{role_omega} and Proposition~\\ref{periodic_solution_prop} it follows immediately\n \n \\begin{Lcor}\n \\label{CorExemploCaos}\nFor small $\\mu>0$, $\\nu>0$ and  for $\\omega>0$ large enough,  the flow of \\eqref{general4} exhibits a hyperbolic rotational horseshoe\nand for a set of positive Lebesgue measure of parameters it also contains  strange attractors. \n \\end{Lcor}\n \n \n\\section{Local coordinates and first return map}\n\\label{secPrelim}\n\n\nIn this section we will analyse the dynamics near the  heteroclinic attractor $\\Gamma$ through local maps, after selecting appropriate coordinates near the saddles   ${\\mathcal P}_{\\mathbf v}=\\{{\\mathbf v}\\}\\times{\\mathbf S}^1$ and  ${\\mathcal P}_{\\mathbf w}=\\{{\\mathbf w}\\}\\times{\\mathbf S}^1$.\n\\subsection{Geometry near ${\\mathcal P}_{{\\mathbf v}}$ and ${\\mathcal P}_{{\\mathbf w}}$}\n\\label{Local}\nLet $U_a$\nbe pairwise disjoint compact neighbourhoods in ${\\mathcal M}^3$ of the nodes  ${\\mathcal P}_{a}$, $a\\in \\{{\\mathbf v},{\\mathbf w}\\}$, such that each boundary  $\\partial U_a$ is a finite union of smooth surfaces delimited by curves, each surface\n transverse to the vector field everywhere, except at its\n boundary.  \nEach  $U_a$ is called an \\emph{isolating block} for  ${\\mathcal P}_{a}$ and, topologically, it consists of a hollow cylinder.  \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=9cm]{local_map.pdf}\n\\end{center}\n\\caption{\\small Top:  isolating block near the periodic solution ${\\mathcal P}_a$, $a\\in \\{{\\mathbf v},{\\mathbf w}\\}$.\nBottom: coordinates at the $In$ and $Out$ components of the boundary.  \nDouble bars mean that the sides are identified.    }\n\\label{local_map_LR_torus1}\n\\end{figure}\n\n\nFor $a\\in \\{{\\mathbf v},{\\mathbf w}\\}$, let $\\Sigma_a$ be a cross section to the flow at $p_a \\in {\\mathcal P}_{a}$. Since ${\\mathcal P}_{a}$ is hyperbolic, there is a neighbourhood $U^*_a$ of $p_a$ in $\\Sigma_a$ where the first return map to $\\Sigma_a$\nis $C^1$ conjugate to its linear part. \n Let $e^{-c_a}$ and $e^{e_a}$, with $c_a,e_a>0$,  be the eigenvalues\nof the derivative $D{\\mathcal F}_{(\\nu, \\mu)}(a)$.\nThen, for each $k\\ge 2$ there is an open and dense subset of ${\\mathbf R}^2$ such that, if \n$(-c_a,e_a)$\n\n lies\n in this set, then the conjugacy is of class $C^k$ (details may be checked in  Appendix A of \\cite{LR17}).\n\nSuspending the linear map gives rise, in cylindrical coordinates $(\\rho, \\theta, z)$ around ${\\mathcal P}_{a}$, to the  equations\n\\begin{equation}\n\\label{ode of suspension}\n\\left\\{ \n\\begin{array}{l}\n\\dot{\\rho}=-c_{a}(\\rho -1) \\\\ \n\\dot{\\theta}=2\\omega \\\\ \n\\dot{z}=e_{a}z\n\\end{array}\n\\right.\n\\end{equation}\n whose flow is\n$C^2$-conjugate to the original flow near ${\\mathcal P}_{a}$. In these coordinates, the periodic trajectory ${\\mathcal P}_{a}$ is the circle defined by $\\rho=1$ and $z=0$.\nFor the moment, let $W^s_{loc}({\\mathcal P}_{a})$ and $W^u_{loc}({\\mathcal P}_{a})$ be the connected components of $W^s({\\mathcal P}_{a})$ and $W^u({\\mathcal P}_{a})$, respectively, contained in the suspension of $U^*_a$  and containing ${\\mathcal P}_a$ in their closure.\nIn these coordinates,  $W^s_{loc}({\\mathcal P}_{a})$, is  the plane $z=0$ and   $W^u_{loc}({\\mathcal P}_{a})$ is the surface $\\rho=1$.\n\n\nAs illustrated in Figure \\ref{local_map_LR_torus1}, we consider a  hollow three-dimensional cylinder $V_a(\\varepsilon_a)$  of \n${\\mathcal P}_{a}$ contained in the suspension of $U^*_a$ \n(with small $\\varepsilon_a>0$ to be determined later)\ngiven by\n$$\nV_a(\\varepsilon_a)=\\left\\{ (\\rho,\\theta,z):\\quad 1\\le\\rho< 1+\\varepsilon_a,\n\\quad 0\\le z< \\varepsilon_a\\quad \\text{and}\\quad \n\\theta\\in{\\mathbf R}\\pmod{2\\pi}\n\\right\\}\\  .\n$$\nWhen there is no ambiguity, we write $V_a$ instead of $V_a(\\varepsilon_a)$.\nIts boundary contains the trajectory ${\\mathcal P}_{a}$ and is a \nunion\n$$\n\\partial V_{a}= In({\\mathcal P}_{a}) \\cup Out({\\mathcal P}_{a}) \\cup \\mathcal{W} ({\\mathcal P}_{a})\n$$\nwhere \n\\begin{itemize}\n\\item\n$\\mathcal{W} ({\\mathcal P}_{a})={\\mathcal P}_{a}\\cup  \\left(W^s_{loc}({\\mathcal P}_{a})\\cap V_a\\right) \\cup  \\left(W^u_{loc}({\\mathcal P}_{a})\\cap V_a\\right)$.\n\\item\n$ W^s_{loc}({\\mathcal P}_{a})\\cap V_a$ is the lower boundary of the hollow cylinder, given by $z=0$, $1<\\rho\\le 1+\\varepsilon_a$.\n\\item\n$W^u_{loc}({\\mathcal P}_{a})\\cap V_a$ is the inner boundary of the hollow cylinder, given by $\\rho=1$, $0<z\\le\\varepsilon_a$.\n\\item \n$In({\\mathcal P}_{a})$ is the outer  wall of the cylinder, defined by $\\rho=1+\\varepsilon_a$, $0\\le z\\le\\varepsilon_a$.\\\\\nTrajectories starting at  $In({\\mathcal P}_{a})$ go inside $V_a$ in small positive time.\n\\item \n$Out({\\mathcal P}_{a})$ is the  top  of the cylinder, the annulus defined by $z=\\varepsilon_a$, $1\\le\\rho\\le 1+\\varepsilon_a$.\\\\\nTrajectories starting at $Out({\\mathcal P}_{a})$ go inside $V_a$ in small negative time. \n\\item \nThe vector field is transverse to $In({\\mathcal P}_{a})\\cup Out({\\mathcal P}_{a})$ except at the circle \n$In({\\mathcal P}_{a})\\cap Out({\\mathcal P}_{a})$.  \n\\end{itemize}\n\nThe cylinder wall $In({\\mathcal P}_{a})$ is\nparametrised by the covering map\n$$\n(\\varphi,r)\\mapsto(1+\\varepsilon_a,\\varphi,r)=(\\rho,\\theta,z),\n$$\nwhere $\\varphi\\in{\\mathbf R}$ and\n$0\\le r\\le\\varepsilon_a$. \nThe annulus $Out({\\mathcal P}_{a})$ is parametrised by the covering\n$$\n(\\varphi,r) \\mapsto ( r,\\varphi, \\varepsilon_a)=(\\rho,\\theta,z),\n$$\nfor $1\\le r\\le 1+\\varepsilon_a$ and $\\varphi\\in{\\mathbf R}$.\n\n\nFor $\\nu\\neq 0$,  $\\mu= 0$ we have from \\eqref{L5} of Proposition~\\ref{PropB1} that $W^u({\\mathcal P}_{\\mathbf v}) \\cap W^s({\\mathcal P}_{\\mathbf w})=\\varnothing$ and the same assumption is made for $\\mu\\ne 0$ in Theorem~\\ref{role_omega}.\nSince  for $\\nu=\\mu=0$ these invariant manifolds coincide, then for small $\\mu,\\nu\\ne 0$ the manifold $W^u({\\mathcal P}_{\\mathbf v})$ must come close to ${\\mathcal P}_{\\mathbf w}$.\nWithout loss of generality we may assume that it meets $In({\\mathcal P}_{\\mathbf v})$,\nwe are concerned with the parameter values for which this holds.\n\nFrom now on the portion of the unstable manifold of ${\\mathcal P}_{\\mathbf v}$  that goes from ${\\mathcal P}_{\\mathbf v}$ to $In({\\mathcal P}_{\\mathbf w})$ without intersecting $V_{\\mathbf w}$ will be  denoted $W^u_{loc}({\\mathcal P}_{\\mathbf v})$. Similarly, $W^s_{loc}({\\mathcal P}_{\\mathbf w})$ will denote the portion of the stable manifold of ${\\mathcal P}_{\\mathbf w}$ that is outside $V_{\\mathbf w}$ and goes directly from $Out({\\mathcal P}_{\\mathbf w})$ to ${\\mathcal P}_{\\mathbf v}$ in negative time.\n\n\\subsection{Local and global maps}\\label{sublocal}\nFor each $a\\in\\{{\\mathbf v}, {\\mathbf w}\\}$, we may solve  (\\ref{ode of suspension}) explicitly, then we compute the flight time from  $In({\\mathcal P}_{a})$ to $Out({\\mathcal P}_{a})$ by solving the equation $z(t)=\\varepsilon_a$ for the trajectory whose initial condition is \n$(\\rho,\\theta,z)=(1+\\varepsilon_a,\\varphi,r) \\in In({\\mathcal P}_{a})\\backslash W^s({\\mathcal P}_{a})$, with $z>0$, as in \\cite{LR17}.\n  Replacing this time in the other coordinates of the solution, yields the local map \n  $$\n  \\Phi _{a}:In({\\mathcal P}_{a})\\backslash W^s({\\mathcal P}_{a})\\, \\longrightarrow\\,  Out({\\mathcal P}_a)\n  $$\n  given by\n \\begin{equation}\n\\label{local map}\n\\Phi _{a}(\\varphi,r)=\n\\left(\\varphi-\\frac{2\\omega}{e_a}\\ln\\left(\\frac{r}{\\varepsilon_a}\\right),\n1+ \\varepsilon_a \\left(\\frac{r}{\\varepsilon_a}\\right)^{\\delta_a}\\right) \n \\end{equation}\n where $\\delta_a=\\dfrac{c_{a}}{e_{a}}>0$. \nFor the transition maps from one isolating block to the other we use assumptions \\ref{A8} and \\ref{A9}.\nWith this notation, we formulate them as follows: \n \\begin{itemize}  \n\\item \nThe two sets  $W^u({\\mathcal P}_{{\\mathbf w}})$  and $W^s({\\mathcal P}_{{\\mathbf v}})$ coincide; \n\\item\nThe manifold $W^u_{loc}({\\mathcal P}_{{\\mathbf v}})$ intersects\n the cylinder $In({\\mathcal P}_{{\\mathbf w}})$  on a non-contractible closed curve $\\gamma_{(\\nu, \\mu)}$. \n\\end{itemize}\nWe will assume that  the universal cover of $\\gamma_{(\\nu, \\mu)}$ is the graph of a smooth\nMorse function  \n$$\n\\xi_{(\\nu, \\mu)}:{\\mathbf R} \\rightarrow [0, \\varepsilon_{\\mathbf w}],\n$$\n as in Figure \\ref{xi1n},\nsatisfying the following conditions for $\\nu>0$: \n\\begin{itemize}\n\\item \n$\\xi_{(\\nu, \\mu)}$ is not constant \n (because the perturbing term $\\phi_\\mu$ is not constant)\n and is $2\\pi$-periodic; \n\\item \n$\\xi_{(\\nu, \\mu)}$ has a local maximum at  $\\varphi_1>0$, a local minimum at $\\varphi_2>\\varphi_1$ and no other critical point in the interval $(\\varphi_1,\\varphi_2)$;  \n\\item if $\\mu>0$ and $\\nu>0$ then $\\forall \\varphi\\in  {\\mathbf R}$,  $\\xi_{(\\nu, \\mu)}(\\varphi)>0$;  \n\\item $\\displaystyle \\lim_{\\nu\\rightarrow 0}\\,  \\max_{\\varphi\\in {\\mathbf R} } \\xi_{(\\nu, \\mu)}(\\varphi)=0$. \n\\end{itemize}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=5cm]{xi1nn.pdf}\n\\end{center}\n\\caption{\\small The local unstable manifold of  ${\\mathcal P}_{{\\mathbf v}}$ intersects the cylinder $In({\\mathcal P}_{{\\mathbf w}})$  on a closed curve, the graph of a periodic Morse function  $\\xi_{(\\nu, \\mu)}:{\\mathbf R} \\rightarrow [0, \\varepsilon_{\\mathbf w}]$ with $\\nu>0$, $\\mu>0$. }\n\\label{xi1n}\n\\end{figure}\n\nWith these assumptions, we may take  the maps $$\\Psi_{{\\mathbf v} \\rightarrow {\\mathbf w}}: Out({\\mathcal P}_{\\mathbf v})\\longrightarrow In ({\\mathcal P}_{\\mathbf w}) \\qquad \\text{and} \\qquad \\Psi_{{\\mathbf w} \\rightarrow {\\mathbf v}}: Out({\\mathcal P}_{\\mathbf w})\\longrightarrow In ({\\mathcal P}_{\\mathbf v})$$ to be given by\n\\begin{equation}\\label{transition21}\n\\qquad \\Psi_{{\\mathbf v} \\rightarrow {\\mathbf w}}(\\varphi,r)=\\left(\\, \\varphi, \\,\\, (r-1)+ \\xi_{(\\nu, \\mu)}(\\varphi)\\right) \\qquad\\text{and}\\qquad \\Psi_{{\\mathbf w} \\rightarrow {\\mathbf v}}(\\varphi,r)=\\left(\\, \\varphi, \\,\\, (r-1)\\right).\n\\end{equation}\n\\subsection{The return map}\n Let  ${\\mathcal R}_{(\\nu,  \\mu)}=\\Phi_{{\\mathbf v}} \\circ \\Psi_{{\\mathbf w} \\rightarrow {\\mathbf v}} \\circ \\Phi_{{\\mathbf w}} \\circ \\Psi_{{\\mathbf v} \\rightarrow {\\mathbf w}}$\n  be the first return map to $Out({\\mathcal P}_{\\mathbf v})$, well defined on the set   of initial conditions $(\\varphi,r) \\in Out({\\mathcal P}_{\\mathbf v})$ whose solution returns to $Out({\\mathcal P}_{\\mathbf v})$. \n For  $r>1$, the map ${\\mathcal R}_{(\\nu,\\mu)}$ is given by\n \\begin{eqnarray*}\\label{first1}\n{\\mathcal R}_{(\\nu, \\mu)}(\\varphi,r)&=& \\left[ \n\\varphi-\\omega K\\ln \\left[(r-1)+\\xi_{(\\nu, \\mu)} (\\varphi)\\right] -\\omega k_\\varepsilon\\pmod{2\\pi},\\ \n 1+\\dfrac{\\varepsilon_{\\mathbf v}}{\\varepsilon_{\\mathbf w}^{\\delta_{\\mathbf w}}}\\left[(r-1)+\\xi_{(\\nu, \\mu)} (\\varphi) \\right] ^\\delta\\right]\\\\\n&=& \\left(R_1(\\varphi,r), R_2(\\varphi ,r)\\right)\n\\end{eqnarray*}\nwhere\n$$\n\\delta = \\delta_{\\mathbf v} \\delta_{\\mathbf w}>1, \\qquad  \nK = 2 \\left( \\frac{e_{\\mathbf v} + c_{\\mathbf w}}{e_{\\mathbf v} \\, e_{\\mathbf w}} \\right)>0 \\qquad \\text{and}\\qquad \nk_\\varepsilon =-2K\\ln \\varepsilon_{\\mathbf w}>0.\n$$\nThe map ${\\mathcal R}_{(\\nu, \\mu)}$ is well defined if $\\varepsilon_{\\mathbf v}+\\displaystyle\\max_{0\\le \\varphi\\le 2\\pi}\\xi_{(\\nu, \\mu)} (\\varphi)\\le \\varepsilon_{\\mathbf w}$.\nIn particular, we need $\\varepsilon_{\\mathbf v}< \\varepsilon_{\\mathbf w}$. \n The inequality $\\delta>1$ comes from Property  \\ref{A5} and the conditions  of \\cite{KM1, KM2} for a heteroclinic cycle to be attracting.\n\n\n\\section{Proof of Theorem~\\ref{role_omega}}\n\\label{secProva_G}\n\nThe goal of the section\nis to obtain  an invariant set  $\\Lambda\\subset Out^\\pm({\\mathcal P}_{\\mathbf v})$ where the map ${\\mathcal R}_{(\\nu, \\mu)}|_\\Lambda$ is topologically conjugate to a Bernoulli shift with two symbols. \nThe argument uses the Conley-Moser conditions, see for instance \\cite{Wiggins}.\n\\subsection{Stretching the angular component}\nLet $[\\varphi_L,\\varphi_R]$ be an interval where $\\xi_{(\\nu, \\mu)} (\\varphi)$ is monotonically decreasing, with \n$$\n\\xi_L=\\xi_{(\\nu, \\mu)} (\\varphi_L)>\\xi_{(\\nu, \\mu)} (\\varphi_R)= \\xi_R\n$$\n and consider $\\mathcal{D}\\subset Out({\\mathcal P}_{{\\mathbf v}})$ parametrised by $(\\varphi,r)\\in [\\varphi_L,\\varphi_R]\\times[1,1+\\varepsilon_{\\mathbf v}]$, with $\\varepsilon_{\\mathbf v}+\\xi_L<\\varepsilon_{\\mathbf w}$.\nThen $\\mathcal{D}$ is a set  of initial conditions $(\\varphi,r) \\in Out({\\mathcal P}_{\\mathbf v})$ whose solution returns to $Out({\\mathcal P}_{\\mathbf v})$. We start by establishing some properties of the map ${\\mathcal R} $  within this set.\n\n\\begin{lemma}\n\\label{lema:contract}\nFor small $\\nu,\\mu>0$, the following assertions hold in  $\\mathcal{D}$  with $\\varepsilon_{\\mathbf v}+\\xi_L<\\varepsilon_{\\mathbf w}$: \n\\begin{enumerate}\n\\item\nfor any $r\\in [1,1+\\varepsilon_{\\mathbf v}]$ the map $\\varphi\\to R_1(\\varphi,r)$ is an expansion;\n\\item\nfor any $\\varphi\\in[\\varphi_L,\\varphi_R]$  the  map $r\\to R_2(\\varphi,r)$ is a contraction. \n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof} \nThe first assertion follows from\n$$\n\\dfrac{\\partial  R_1 (\\varphi,r)}{\\partial \\varphi } =\n1-\\dfrac{\\omega K}{r-1+\\xi_{(\\nu, \\mu)}( \\varphi) }\\dfrac{d\\xi_{(\\nu, \\mu)}( \\varphi)}{d\\varphi}>1\n$$\nbecause $\\xi_{(\\nu, \\mu)}( \\varphi) >0$ and  $\\dfrac{d\\xi_{(\\nu, \\mu)}( \\varphi)}{d\\varphi}<0$ since we are assuming $\\xi_{(\\nu, \\mu)}$ is monotonically decreasing in $[\\varphi_L,\\varphi_R]$.\nThe second assertion follows from\n$$\n\\dfrac{\\partial  R_2 (\\varphi,r)}{\\partial r } = \\delta\\dfrac{\\varepsilon_{\\mathbf v}}{\\varepsilon_{\\mathbf w}^{\\delta_{\\mathbf w}}} \\left[(r-1) +\\xi_{(\\nu, \\mu)}( \\varphi)\\right]^{\\delta-1}. \n$$\nSince  $0<\\xi_{(\\nu, \\mu)}( \\varphi)\\le \\xi_L$  and $0<r-1\\leq \\varepsilon_{\\mathbf v}\\le\\varepsilon_{\\mathbf w}<1$ then\n$$\n0<\\frac{\\partial  R_2 (\\varphi,r)}{\\partial r } \n\\le \\delta\\dfrac{\\varepsilon_{\\mathbf v}}{\\varepsilon_{\\mathbf w}^{\\delta_{\\mathbf w}}}\\left[\\varepsilon_{\\mathbf w}+\\xi_L \\right]^{\\delta-1}\n=\\delta\\dfrac{\\varepsilon_{\\mathbf v}}{\\varepsilon_{\\mathbf w}}+\\mathcal{O}(\\xi_L),\n$$\nwhere $\\mathcal{O}$ stands for the standard Landau notation. We have $\\displaystyle\\lim_{\\nu\\rightarrow 0} \\max_{\\varphi\\in[\\varphi_L,\\varphi_R]} \\xi_{(\\nu, \\mu)}(\\varphi)=0$,\ntherefore $0<\\dfrac{\\partial  R_2}{\\partial r} (\\varphi,r) <1 $ for small $\\mu>0$. \n\\end{proof}\n\nWe call the graph $(\\varphi,s(\\varphi))$ in  $\\mathcal{D}$ of a monotonic map $s(\\varphi)$ with \n$\\varphi_L\\le\\varphi\\le\\varphi_R$, a  \\emph{segment across} $\\mathcal{D}$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=4cm]{spiral.pdf} \n\\end{center}\n\\caption{\\small When $\\omega\\ge\\omega_0$, the segment $S$ (red) in the domain  $\\mathcal{D}\\subset Out({\\mathcal P}_{{\\mathbf v}})$ (gray) is  transformed by the first return map ${\\mathcal R} $ into a curve (blue) that makes a full turn around $Out(\\mathcal{P}_{\\mathbf v})$ intersecting $\\mathcal{D}$ in at least one segment.}\n\\label{fig:spiral}\n\\end{figure}\n\n\n\\begin{lemma}\\label{lema:spiral}\nConsider the segment \n$S=\\{(\\varphi,r_*)\\ \\varphi_L\\le\\varphi\\le\\varphi_R \\}\\subset \\mathcal{D}$\nfor a given $r_*>0$.\nFor  small $\\mu,\\nu>0$   with $\\varepsilon_{\\mathbf v}+\\xi_L<\\varepsilon_{\\mathbf w}$, if \n$$\n\\omega\\ge\\omega_0=\\dfrac{ 2\\pi}{K\\ln( 1+(\\xi_L-\\xi_R)\/(1+\\xi_R))}\n$$ \nthen for any $r_*\\in(1,1+\\varepsilon_{\\mathbf v}]$,\nthe set ${\\mathcal R} (S)\\cap \\mathcal{D}$ is a curve   containing a segment across $\\mathcal{D}$.\n\\end{lemma}\n\n\\begin{proof}\nSince  $\\xi_{(\\nu, \\mu)} (\\varphi)$ is monotonically decreasing in $[\\varphi_L,\\varphi_R]$ then  the map \n$ \\varphi\\to R_2(\\varphi, r_*)$ is monotonically decreasing and \n$$ \\varphi\\mapsto \\varphi-\\omega K\\ln \\left[(r-1)+\\xi_{(\\nu, \\mu)} (\\varphi)\\right] -\\omega k_\\varepsilon=R_1(\\varphi,r_*)$$ is monotonically increasing  in the same interval. \nHence ${\\mathcal R} (S)$ is the graph of a monotonic map $s(\\varphi)$, with the map $s$ defined in some interval  $I$, as in Figure~\\ref{fig:spiral}. \nIt remains to obtain an estimate of the variation of the first coordinate of ${\\mathcal R} (S)$ to ensure that $[\\varphi_L,\\varphi_R]\\subset I$.\nFrom the definition of $R_1$ and properties of the logarithm, one knows that the difference $\\Delta =R_1(\\varphi_R,r_*) - R_1( \\varphi_L, r_*)$ satisfies\n$$\n\\begin{array}{rl}\n\\Delta =&\n\\left(\\varphi_R-\\omega K\\ln \\left[(r_*-1)+\\xi_{(\\nu, \\mu)} (\\varphi_R)\\right] -\\omega k_\\varepsilon\\right)-\n\\left(\\varphi_L-\\omega K\\ln \\left[(r_*-1)+\\xi_{(\\nu, \\mu)} (\\varphi_L)\\right] -\\omega k_\\varepsilon\\right)\\\\ \\\\\n=&(\\varphi_R-\\varphi_L)+\\omega K\n \\ln\\dfrac{(r_*-1)+\\xi_{(\\nu, \\mu)} (\\varphi_L)}{(r_*-1)+\\xi_{(\\nu, \\mu)} (\\varphi_R)}\\\\ \\\\\n =&(\\varphi_R-\\varphi_L)+\\omega K\n  \\ln \\dfrac{(r_*-1)+\\xi_L}{(r_*-1)+\\xi_R}> (\\varphi_R-\\varphi_L)\n\\end{array}\n$$\nwhere for the last inequality we use $\\xi_L>\\xi_R$ hence $ \\ln \\dfrac{(r_*-1)+\\xi_L}{(r_*-1)+\\xi_R}>0$.\nMoreover,\n$$\n\\dfrac{(r_*-1)+\\xi_L}{(r_*-1)+\\xi_R}=\n1+\\dfrac{\\xi_L-\\xi_R}{(r_*-1)+\\xi_R}\\ge \n1+\\dfrac{\\xi_L-\\xi_R}{1+\\xi_R}.\n$$\nTherefore, if $\\omega\\ge\\dfrac{ 2\\pi}{K\\ln( 1+(\\xi_L-\\xi_R)\/(1+\\xi_R))}$, then $\\Delta \\ge 2\\pi+  (\\varphi_R-\\varphi_L)$ and hence the curve ${\\mathcal R} (S)$ goes across $\\mathcal{D}$ at least once, as in Figures~\\ref{fig:spiral} and \\ref{horseshoe_fig}.\n\\end{proof}\n\n\\subsection{Proof of Theorem~\\ref{role_omega}. Part I}\n\\label{prova_parte1}\nGiven a rectangular region   in $Out({\\mathcal P}_{\\mathbf v})$, parametrised by \n$ [\\varphi_a, \\varphi_b]\\times [r_1,r_2]$, a \\emph{vertical  strip} in the region is a set\n$$\n{\\mathcal V}=\\{(\\varphi,r): \\varphi\\in[u_1(r),u_2(r)]\\qquad r \\in \\,[r_1,r_2]\\}\n$$\nwhere $u_1,u_2: [r_1,r_2] \\rightarrow [\\varphi_a,\\varphi_b]$ are Lipschitz functions with Lipschitz constants less than $\\mu_v\\ge 0$, such that $u_1(r)<u_2(r)$. The \\emph{ vertical  boundaries} of a  vertical  strip are the graphs of the maps $u_i$; the \\emph{horizontal boundaries} are the lines $\\{r_i\\} \\times  [u_1(r_i),u_2(r_i)]$, $i=1,2$; \nthe \\emph{width} $d({\\mathcal V})$ of the strip is \n$$\nd({\\mathcal V})=\\max_{r_1\\le r\\le r_2} |u_1(r)-u_2(r)|.\n$$\nIn an analogous way we define a \\emph{horizontal strip} across  a \\emph{horizontal rectangle} in $Out({\\mathcal P}_{\\mathbf v})$ with the roles of $\\varphi$ and $r$ reversed, Lipschitz constants less than $\\mu_h\\ge 0$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=6cm]{stretching2_LR_torus.pdf}\n\\end{center}\n\\caption{\\small  For $r_\\star \\in  [0, 1]$, the image under ${\\mathcal R}$ of the segment $[\\varphi_L, \\varphi_R]\\times \\{r=r_* \\}$ is a curve (without folds) intersecting twice the rectangle $\\mathcal{D}$. \nThe  subsets $I_1$ and $I_2$ are stretched by the first return map ${\\mathcal R}$ into segments across $\\mathcal{D}$. }\\label{horseshoe_fig}\n\\end{figure}\n\n\nLet $I_1=[\\varphi_a, \\varphi_b]$ and $I_2=[\\varphi_c, \\varphi_d]$ be two disjoint intervals satisfying\n$$\n\\varphi_L\\le\\varphi_a<\\varphi_b<\\varphi_c<\\varphi_d\\le \\varphi_R.\n$$\nWe claim that for $\\omega\\ge\\omega_0$ (of Lemma~\\ref{lema:spiral}) the  two  vertical  strips\n$$\n{\\mathcal V}_i= \\{(\\varphi,r)\\in \\mathcal{D}: \\varphi\\in I_i\\}\\qquad i=1,2\n$$\nsatisfy the Conley-Moser conditions \\cite{Wiggins}: \n\\begin{enumerate}\n\\renewcommand{\\theenumi}{(P\\arabic{enumi})}\n\\renewcommand{\\labelenumi}{{\\theenumi}}\n\\item\\label{f(V)=H}\nThe image ${\\mathcal R} (V_i)\\cap \\mathcal{D}=H_i$ is the union of disjoint horizontal strips across $\\mathcal{D}$.\n\\item\\label{f(V)capV}\nFor every vertical strip $V\\subset (V_1\\cup V_2)$ the set ${\\mathcal R} ^{-1}(V)\\cap V_i=\\widetilde{V_i}$ is a vertical strip across $\\mathcal{D}$ with $d\\left(\\widetilde{V_i}\\right)\\le \\lambda_v d(V)$ for some $\\lambda_v\\in(0,1)$.\n\\item\\label{f(H)capH}\nFor every horizontal strip $H\\subset (H_1\\cup H_2)$ the set ${\\mathcal R} (H)\\cap H_i=\\widetilde{H_i}$ is a horizontal strip across $\\mathcal{D}$ with $d\\left(\\widetilde{H_i}\\right)\\le \\lambda_h d(H)$ for some $\\lambda_h\\in(0,1)$.\n\\end{enumerate}\nIn order to establish the claim, note that by Lemma~\\ref{lema:spiral} each line with constant $r$ in $\\mathcal{V}_i$ is mapped by ${\\mathcal R}$ into a curve in $Out(\\mathcal{P}_{\\mathbf v})$ that intersects $\\mathcal{D}$ in at least one segment. \n Lemma~\\ref{lema:contract} ensures that as $r$ varies in $]1,1+\\varepsilon_{\\mathbf v}]$, the second coordinate of their image varies in an interval of length less than $\\varepsilon_{\\mathbf v}$, so the union of these segments lies in horizontal strips across $\\mathcal{D}$, establishing \\ref{f(V)=H}.\nProperties~\\ref{f(V)capV} and  \\ref{f(H)capH} follow from Lemma~\\ref{lema:contract}.\nFrom this claim it follows that there exists an ${\\mathcal R}$-invariant set of initial conditions \n$$\n{\\Lambda}=\\bigcap_{n \\in {\\mathbf Z}} {\\mathcal R}^n({\\mathcal V}_1 \\cup {\\mathcal V}_2)\n$$ \non which $\\left.{\\mathcal R}\\right|_\\Lambda$ is topologically conjugate to a Bernoulli shift on two symbols. By construction it is a rotational horseshoe (according to \\cite{Passeggi}) with $m=2$.\n \n \n\n \n\n\n\\subsection{Proof of Theorem~\\ref{role_omega}. Part II}\nFor $\\nu>0$ and $\\mu=0$, the flow of \\eqref{general} has an attracting two-dimensional torus (by Proposition \\ref{PropB1}). In particular, there is a cross section $\\Sigma$ where the torus defines an invariant curve $\\mathcal{C}$ under the first return map ${\\mathcal R}$.  Furthermore, there is countable set of values of the type $(\\nu_i, 0)$, $i \\in {\\mathbf N}$, for which the first return map ${\\mathcal R} $  has at least one saddle and a sink lying on $\\mathcal{C}$ ($\\Rightarrow$ the torus is decomposed into periodic orbits with rational rotation number). Fix, once for all, one of these values.\n\nFor such a $\\nu_i>0$, increasing $\\mu>0$ the coexistence of this pair of periodic orbits persists along a wedge, the so called \\emph{Arnold tongue} \\cite{Anishchenko}. As illustrated in Figure \\ref{Strange_attractors1_LR_torus}, for a fixed $\\mu>0$, we know that:\n\\begin{itemize}\n\\item ${\\mathcal R}(\\mathcal{C})$ is a closed curve on $Out^+({\\mathcal P}_{\\mathbf v})$ because ${\\mathcal R}|_\\mathcal{D}$ is a diffeomorphism;\n\\item for $\\omega \\approx 0$,  this curve may be seen as the graph on $Out^+({\\mathcal P}_{\\mathbf v})$ of a non-constant map  defined on $[0, 2\\pi]$ (cf. \\cite{Rodrigues2019}); the curve $\\mathcal{C}$ is the $\\omega$-limit of  $W^u({\\mathcal P}_{\\mathbf w})$;\n\\item  by Lemma \\ref{lema:spiral}, there exists $\\omega_0>0$ and a segment $S\\subset \\mathcal{C}$ such that ${\\mathcal R} (S)\\cap \\mathcal{D}$ is a curve containing a segment across $\\mathcal{D}$.\n\\end{itemize}\n\nThis means that the  curve  $\\mathcal{C}$ starts to develop folds as in Figure~\\ref{Strange_attractors1_LR_torus} (B) and (C). \nIf $\\omega>\\omega_0$  it creates the rotational horseshoes proved in \\S \\ref{prova_parte1}.\n Within this wedge, \nAnishchenko, Safonova and Chua have shown in \\cite{Anishchenko}\n that there are curves on the parameter space $(\\nu, \\mu)$ corresponding to a quadratic homoclinic tangency associated to a dissipative periodic point of the first return map ${\\mathcal R}$.   \n\nUsing now the results of Mora and Viana in  \\cite{MV93}, there exists a positive measure set $\\mathcal{U}$ of parameter values, so that for every $(\\nu, \\mu) \\in \\mathcal{U}$, the return map $\\mathcal{R}$ \nadmits a strange attractor  of H\\'enon-type. \nThese strange attractors are supported in SRB measures. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12cm]{Strange_attractors1_LR_torus.pdf}\n\\end{center}\n\\caption{\\small  Image of $ {\\mathcal R}(\\mathcal{C})$ for different values of $\\omega$ with  $\\nu>0$ and $\\mu$ fixed.\nTransition from an invariant and attracting curve (A) to a rotational horseshoe (C), passing through a homoclinic tangency (B). One observes the breaking of the wave which accompanies the break of the invariant circle  (corresponding to the torus).  Here the point F is a sink and the point S is a saddle. In (C), a neighborhood of $r = 1$ is folded and mapped into itself, leading to the formation of rotational horseshoes. }\n\\label{Strange_attractors1_LR_torus}\n\\end{figure}\n\n\n\n\\begin{remark}\nIn this type of result, the number of connected components with which the strange attractors intersect the section $\\Sigma$ is not specified nor is the size of their basins of attraction. The strange attractors coexist with sinks from Newhouse phenomena. A discussion of these results may be found in \\cite{CR2021, Rodrigues2019, TS86}. \n\\end{remark}\n\n\n\n\n\n\\section{Proof of Proposition \\ref{periodic_solution_prop} }\n\\label{PropA}\n\n\n\nThis proposition concerns the case $\\mu=0$ when the time-periodic perturbation to $\\dot x={\\mathcal F}_\\nu(x)$ is constant.\nFor $\\nu=0$, Properties~\\ref{A1} to \\ref{A4} and \\ref{A6} of $\\dot x={\\mathcal F}_0(x)$ were established in  \\cite[Theorem 7]{ACL06}, with ${\\mathcal M}^2={\\mathbf S}^2$.\n \n\\begin{proof} The $\\kappa$-equivariance is easily checked directly from the expression of ${\\mathcal F}_\\nu$.\n\n The first part of the proof of Proposition \\ref{periodic_solution_prop}  consists in establishing that Properties~\\ref{A1} to \\ref{A3}  persist when the perturbation term $\\nu(1-x_1)$ is added.\n Properties~\\ref{A4} and \\ref{A6} are established in \\cite{ACL06} and Property~\\ref{A5} is a consequence of their results.\nThen it remains to show that two periodic solutions are created by the perturbation when the connections from ${\\mathbf w}$ to ${\\mathbf v}$ are broken, as stated in \\ref{A7}.\nAddressing the  persistence and property  \\ref{A7} constitutes the remainder of this proof.\n\n\\begin{itemize}\n\\item[\\ref{A1}] \nSince for $\\nu=0$ the sphere ${\\mathbf S}^2$ is normally hyperbolic as in  \\cite{HPS}, then for small $\\nu\\ne 0$ it persists as a flow-invariant, normally hyperbolic, globally attracting manifold ${\\mathcal M}^2$. See also the analysis by \\cite{HG97}.  \n\n\\item[\\ref{A2}] \nFor  $\\nu=0$ the only equilibria in the flow-invariant plane $\\Fix({\\mathbf Z}_2(\\kappa))$ are ${\\mathbf v}$ and ${\\mathbf w}$ above, as well as  the origin $O$.\nSince they are hyperbolic, then their hyperbolic continuations should exist within the plane $\\Fix({\\mathbf Z}_2(\\kappa))$.\nAnother way to see this is to  solve\n$$\n\\left\\{\\begin{array}{l}\nF_1(x_1,x_3)= x_1[(1-x_1^2-x_3^2)-\\alpha  x_3 +\\beta x_3^2 - \\nu] + \\nu =0\\\\ \\\\\nF_2(x_1,x_3)=  x_3[(-x_1^2-x_3^2) -\\beta x_1^2] +\\alpha x_1^2=0 .\n\\end{array}\\right.\n$$\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=75mm]{G1_G2_1A.pdf}\\qquad \\includegraphics[width=75mm]{G1_G2_2n.pdf}\n\\end{center}\n\\caption{\\small Equilibria of $\\dot\\zeta={\\mathcal F}_{\\nu}(\\zeta)$ for $\\nu=0$ (left) and $\\nu=0.5$ (right), $\\alpha=1$, $\\beta=-0.1$ in the flow-invariant subspace $\\Fix({\\mathbf Z}_2(\\kappa))$ occur at the intersection of the curves $F_1(x_1,x_3)=0$ (blue) and $F_2(x_1,x_3)=0$ (red), here plotted with Maxima.\n}\n\\label{G1G2}\n\\end{figure}\n\nFigure~\\ref{G1G2} shows the curves $F_1=0$ and $F_2=0$ plotted with Maxima. \nFor $\\nu=0$ the curves intersect transversely at $O$, ${\\mathbf v}$ and ${\\mathbf w}$ and this property is preserved for small $\\nu\\ne 0$.\n\\item[\\ref{A3}] \nWithin the flow-invariant plane $\\Fix({\\mathbf Z}_2(\\kappa))$, the origin is a repelling source,  ${\\mathbf v}$ is a sink and ${\\mathbf w}$  is a saddle. Since both the plane $y=0$ and the manifold ${\\mathcal M}^2$ are flow-invariant, this means that there are two heteroclinic connections from \n$\\tilde{{\\mathbf w}}$ to $\\tilde{{\\mathbf v}}$.   \n\n\n\\item[\\ref{A5}] \nIn \\cite{ACL06} it is established that the only other equilibria in ${\\mathbf S}^2$ are the four hyperbolic repelling foci $(\\pm \\sqrt{2}\/2, \\pm \\sqrt{2}\/2, 0)$. \nThis means that for  $\\nu=0$, by the Poincar\\'e-Bendixon Theorem,  the $\\omega$-limit set of all other points in ${\\mathbf S}^2$ must be contained in the heteroclinic cycles that contain ${\\mathbf v}$ and ${\\mathbf w}$.\nThe unstable foci  remain for $\\nu\\ne 0$ small. \n\n\\item[\\ref{A7}]\nThe flow-invariant subspace $\\Fix({\\mathbf Z}_2(\\kappa))$ divides ${\\mathcal M}^2$ in two flow-invariant components.\nWe will show that the $x_2>0$ component contains a non-constant periodic solution, the proof for $x_2<0$ follows from the symmetry.\nFor $x_1=0$ and $\\mu=0$  the expression \\eqref{general4} yields $\\dot x=\\nu\\ne 0$.\nSuppose $\\nu>0$, then the region $x_1>0$, $x_2>0$ in ${\\mathcal M}^2$ is positively invariant, see Figure~\\ref{G1G2_5A}.\nThis region only contains one equilibrium, one of the repelling foci in \\ref{A4}, hence by the Poincar\\'e-Bendixon Theorem,\nthe $\\omega$-limit of the unstable manifold of $\\tilde{\\mathbf v}$ is an attracting periodic solution. \nWhen $\\nu<0$  the periodic solution appears  for $x_1<0$.   \nThe period tends to $+\\infty$ as $\\nu$ goes to $0$, since the periodic trajectory  accumulates on\n the heteroclinic cycle.\n\\end{itemize}\n\\end{proof}\n\nProposition \\ref{periodic_solution_prop} shows that for sufficienly small $|\\nu|>0$, each heteroclinic cycle that occurred in the fully symmetric case  \nis replaced by a stable hyperbolic periodic solution. Using the reflection symmetry ${\\mathbf Z}_2(\\kappa)$, two stable periodic solutions co-exist, one in each connected component of $\\mathcal{M}\\backslash \\Fix({\\mathbf Z}_2(\\kappa))$. Their period tends to $\\infty$ when $\\nu$ vanishes  and their basin of attraction must contain the basin of attraction of $\\Sigma_0$. \n \n\n\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=65mm]{G1_G2_5An.pdf}\\includegraphics[width=65mm]{G1_G2_3A.pdf} \\\\\\includegraphics[width=65mm]{G1_G2_4A.pdf}\n\\end{center}\n\\caption{\\small Qualitative phase portrait for the dynamics or \\eqref{general4} in ${\\mathcal M}^2$ with $y>0$, projected   into the $(x_1,0,x_3)$ plane for  $\\mu=0$ and $\\nu>0$ . }\n\\label{G1G2_5A}\n\\end{figure}\n\n\n\\section{Discussion and concluding remarks}\n\\label{discussion}\n\n    \\emph{Routes  to chaos}  have been a recurrent concern on nonlinear dynamics during the last decades \\cite{AS91}. The novelty of the present paper is the illustration of a \\emph{new route} for the emergence of strange attractors  from an attracting heteroclinic network as a codimension-two phenomenon. \n\n\\subsection{Literature}\nIn \\cite{Kaneko}  Kaneko investigates numerically the\n bifurcations of tori in a two-parameter family of dissipative coupled maps. Each map undergoes a period-doubling cascade accumulating on a given parameter value, which generates chaos in the coupled system.   In the same setting Bakri and Verhulst show in  \\cite{Bakri}  that,  for small amplitudes, zero damping and zero coupling, their system reveals a periodic solution which undergoes a Hopf bifurcation, generating an attracting torus. \n They use numerical bifurcation techniques to show   how the torus gets destroyed by dynamical and topological changes in the involved manifolds. The results agree with \\cite{Ruelle, TS86}.\n\nIn the context of dissipative vortex  dynamics Fleurantin and James have studied in \\cite{FJ2020} the  \\emph{Langford system}, a   one-parameter family of three-dimensional vector fields.\nThe flow of this model exhibits a sink, two saddle-foci of different Morse indices and a non-trivial periodic solution with a complex conjugate pair of Floquet exponents. The frequency of the periodic solution together with the frequency of the complex exponent constitute two competing natural modes of oscillation. \nThe periodic orbit undergoes a \nbifurcation  giving rise to observable chaos  through the same mechanism of \\cite{Bakri}.\n These authors studied the evolution of the torus,  its loss of differentiability,  and the appearance of a strange attractor via the existence of  tangencies.  \n  The relative position of the manifolds according to the parameter allows the authors to prove the existence of \\emph{bistability} between an equilibrium and a torus.  \n\n\\subsection{Heteroclinic tangle}\nThe formation of the horseshoe of Theorem \\ref{role_omega} has a different nature to   those found in \\cite{ACL06, LR17} -- in this case, the shift dynamics is obtained via the transverse intersection of two two-dimensional invariant manifolds.  The parameter $\\omega$ is not necessary to prove the existence of chaos.  The non-wandering set associated to the network contains, but does not coincide with, the suspension of horseshoes; it contains  infinitely many heteroclinic pulses and attracting limit cycles with long  periods, coexisting with sets with positive entropy, giving rise to the so called \\emph{quasi-stochastic attractors} \\cite{LR17}.  The sinks  have long periods and narrow basins of attraction.\n\n\n\\subsection{Open questions}\nIn the context of this class of examples, some problems remain to be solved:\n\\begin{enumerate}\n\\item the basins of attraction of the strange attractors of Theorem B are relatively small in terms of Lebesgue measure (they are close to Newhouse domains). Could we improve Theorem B in order to get the existence of a ``larger'' strange attractor?\n\\item is it possible to generalize our result for clean heteroclinic networks (networks whose unstable manifolds are contained within it) whose connections are one-dimensional?\n\\end{enumerate}\nWe believe that these problems can be tackled by using the theory of rank-one attractors developed by Wang and Young \\cite{WY}. We defer these tasks for future work.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}