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In the Stanley lattice defined on Dyck paths of size $n$, cover relations are obtained by replacing a valley $DU$ by a peak $UD$. We investigate a greedy version of this lattice, first introduced by Chenevi\`ere, where cover relations replace a factor $DU^k D$ by $U^kD^2$. By relating this poset to another poset recently defined by Nadeau and Tewari, we prove that this still yields a lattice, which we call the ascent lattice $L_n$. We then count intervals in $L_n$. Their generating function is found to be algebraic of degree $3$. The proof is based on a recursive decomposition of intervals involving two catalytic parameters. The solution of the corresponding functional equation is inspired by recent work on the enumeration of walks confined to a quadrant. We also consider the order induced in $L_{mn}$ on $m$-Dyck paths, that is, paths in which all ascent lengths are multiples of $m$, and on mirrored $m$-Dyck paths, in which all descent lengths are multiples of $m$. The first poset $L_{m,n}$ is still a lattice for any $m$, while the second poset $L'_{m,n}$ is only a join semilattice when $m>1$. In both cases, the enumeration of intervals is still described by an equation in two catalytic variables. Interesting connections arise with the sylvester congruence of Hivert, Novelli and Thibon, and again with walks confined to a quadrant. We combine the latter connection with probabilistic results to give asymptotic estimates of the number of intervals in both $L_{m,n}$ and $L'_{m,n}$. Their form implies that the generating functions of intervals are no longer algebraic, nor even D-finite, when $m>1$., Comment: 35 pages

We consider a greedy version of the $m$-Tamari order defined on $m$-Dyck paths, recently introduced by Dermenjian. Inspired by intriguing connections between intervals in the ordinary 1-Tamari order and planar triangulations, and more generally by the existence of simple formulas counting intervals in the ordinary $m$-Tamari orders, we investigate the number of intervals in the greedy order on $m$-Dyck paths of fixed size. We find again a simple formula, which also counts certain planar maps (of prescribed size) called $(m+1)$-constellations. For instance, when $m=1$ the number of intervals in the greedy order on 1-Dyck paths of length $2n$ is proved to be $\frac{3\cdot 2^{n-1}}{(n+1)(n+2)} \binom{2n}{n}$, which is also the number of bipartite maps with $n$ edges. Our approach is recursive, and uses a ``catalytic'' parameter, namely the length of the final descent of the upper path of the interval. The resulting bivariate generating function is algebraic for all $m$. We show that the same approach can be used to count intervals in the ordinary $m$-Tamari lattices as well. We thus recover the earlier result of the first author, Fusy and Pr\'eville-Ratelle, who were using a different catalytic parameter., Comment: 24 pages

In the past 20 years, the enumeration of plane lattice walks confined to a convex cone -- normalized into the first quadrant -- has received a lot of attention, stimulated the development of several original approaches, and led to a rich collection of results. Most of them deal with the nature of the associated generating function: for which models is it algebraic, D-finite, D-algebraic? By model, what we mean is a finite collection of allowed steps. More recently, similar questions have been raised for non-convex cones, typically the three-quadrant cone $C = \{ (i,j) : i \geq 0 \text{ or } j \geq 0 \}$. They turn out to be more difficult than their quadrant counterparts. In this paper, we investigate a collection of eight models in $C$. This collection consists of diagonally symmetric models in $\{-1, 0,1\}^2\setminus\{(-1,1), (1,-1)\}$. Three of them are known not to be D-algebraic. We show that the remaining five can be solved in a uniform fashion using Tutte's notion of \emph{invariants}, which has already proved useful for some quadrant models. Three models are found to be algebraic, one is (only) D-finite, and the last one is (only) D-algebraic. We also solve in the same fashion the diagonal model $\{ \nearrow, \nwarrow, \swarrow, \searrow\}$, which is D-finite. The three algebraic models are those of the Kreweras trilogy, $S=\{\nearrow, \leftarrow, \downarrow\}$, $S'=\{\rightarrow, \uparrow, \swarrow\}$, and $S\cup S'$. Our solutions take similar forms for all six models. Roughly speaking, the square of the generating function of three-quadrant walks with steps in $S$ is an explicit rational function in the quadrant generating function with steps in $\{(j-i,j): (i,j) \in S\}$. We derive various corollaries, including an explicit algebraic description of the positive harmonic function in $C$ for the five models that are at least D-finite., Comment: 56 pages

We continue the enumeration of plane lattice walks with small steps avoiding the negative quadrant, initiated by the first author in 2016. We solve in detail a new case, namely the king model where all eight nearest neighbour steps are allowed. The associated generating function is proved to be the sum of a simple, explicit D-finite series (related to the number of walks confined to the first quadrant), and an algebraic one. This was already the case for the two models solved by the first author in 2016. The principle of the approach is also the same, but challenging theoretical and computational difficulties arise as we now handle algebraic series of larger degree. We expect a similar algebraicity phenomenon to hold for the seven Weyl step sets, which are those for which walks confined to the first quadrant can be counted using the reflection principle. With this paper, this is now proved for three of them. For the remaining four, we predict the D-finite part of the solution, and in three of the four cases, give evidence for the algebraicity of the remaining part., Comment: 48 pages, 6 figures, 6 tables; full version of the extended abstract arXiv:2109.14307 that appeared in the proceedings of the Analysis of Algorithms Conference 2020

We continue the enumeration of plane lattice paths avoiding the negative quadrant initiated by the first author in [Bousquet-M{\'e}lou, 2016]. We solve in detail a new case, the king walks, where all 8 nearest neighbour steps are allowed. As in the two cases solved in [Bousquet-M{\'e}lou, 2016], the associated generating function is proved to differ from a simple, explicit D-finite series (related to the enumeration of walks confined to the first quadrant) by an algebraic one. The principle of the approach is the same as in [Bousquet-M{\'e}lou, 2016], but challenging theoretical and computational difficulties arise as we now handle algebraic series of larger degree. We also explain why we expect the observed algebraicity phenomenon to persist for 4 more models, for which the quadrant problem is solvable using the reflection principle., Comment: Analysis of Algorithms, 2020, Klagenfurt, Austria

We consider the classical problem of determining the stationary distribution of the semimartingale reflected Brownian motion (SRBM) in a two-dimensional wedge. Under standard assumptions on the parameters of the model (opening of the wedge, angles of the reflections, drift), we study the algebraic and differential nature of the Laplace transform of this stationary distribution. We derive necessary and sufficient conditions for this Laplace transform to be rational, algebraic, differentially finite or more generally differentially algebraic. These conditions are explicit linear dependencies between the angles of the model. A complicated integral expression for this Laplace transform has recently been obtained by two authors of this paper. In the differentially algebraic case, we provide a simple, explicit integral-free expression in terms of a hypergeometric function. It specializes to earlier expressions in several classical cases: the skew-symmetric case, the orthogonal reflections case and the sum-of-exponential densities case (corresponding to the so-called Dieker-Moriarty conditions on the parameters). This paper thus closes, in a sense, the quest of all ``simple'' cases. To prove these results, we start from a functional equation that the Laplace transform satisfies, to which we apply tools from diverse horizons. To establish differential algebraicity, a key ingredient is Tutte's invariant approach, which originates in enumerative combinatorics. It allows us to express the Laplace transform (or its square) as a rational function of a certain canonical invariant, a hypergeometric function in our context. To establish differential transcendence, we turn the functional equation into a difference equation and apply Galoisian results on the nature of the solutions to such equations., Comment: 66 pages

We present recent results on the enumeration of $q$-coloured planar maps, where each monochromatic edge carries a weight $\nu$. This is equivalent to weighting each map by its Tutte polynomial, or to solving the $q$-state Potts model on random planar maps. The associated generating function, obtained by Olivier Bernardi and the author, is differentially algebraic. That is, it satisfies a (non-linear) differential equation. The starting point of this result is a functional equation written by Tutte in 1971, which translates into enumerative terms a simple recursive description of planar maps. The proof follows and adapts Tutte's solution of properly $q$-coloured triangulations (1973-1984). We put this work in perspective with the much better understood enumeration of families of uncoloured planar maps, for which the recursive approach almost systematically yields algebraic generating functions. In the past 15 years, these algebraicity properties have been explained combinatorially by illuminating bijections between maps and families of plane trees. We survey both approaches, recursive and bijective. Comparing the coloured and uncoloured results raises the question of designing bijections for coloured maps. No complete bijective solution exists at the moment, but we present bijections for certain specialisations of the general problem. We also show that for these specialisations, Tutte's functional equation is much easier to solve that in the general case. We conclude with some open questions., Comment: 40 p

Bipolar orientations of planar maps have recently attracted some interest in combinatorics, probability theory and theoretical physics. Plane bipolar orientations with $n$ edges are known to be counted by the $n$th Baxter number $b(n)$, which can be defined by a linear recurrence relation with polynomial coefficients. Equivalently, the associated generating function $\sum_n b(n)t^n$ is D-finite. In this paper, we address a much refined enumeration problem, where we record for every $r$ the number of faces of degree $r$. When these degrees are bounded, we show that the associated generating function is given as the constant term of a multivariate rational series, and thus is still D-finite. We also provide detailed asymptotic estimates for the corresponding numbers. The methods used earlier to count all plane bipolar orientations, regardless of their face degrees, do not generalize easily to record face degrees. Instead, we start from a recent bijection, due to Kenyon et al., that sends bipolar orientations onto certain lattice walks confined to the first quadrant. Due to this bijection, the study of bipolar orientations meets the study of walks confined to a cone, which has been extremely active in the past 15 years. Some of our proofs rely on recent developments in this field, while others are purely bijective. Our asymptotic results also involve probabilistic arguments., Comment: 64 pages. Special issue of the S\'eminaire Lotharingien de Combinatoire, dedicated to Christian Krattenthaler's 60th birthday

We address the enumeration of planar 4-valent maps equipped with an Eulerian orientation by two different methods, and compare the solutions we thus obtain. With the first method we enumerate these orientations as well as a restricted class which we show to be in bijection with general Eulerian orientations. The second method, based on the work of Kostov, allows us to enumerate these 4-valent orientations with a weight on some vertices, corresponding to the six vertex model. We prove that this result generalises both results obtained using the first method, although the equivalence is not immediately clear., Comment: accepted for the conference proceedings of FPSAC 2019

In the past fifteen years, the enumeration of lattice walks with steps takenin a prescribed set S and confined to a given cone, especially the firstquadrant of the plane, has been intensely studied. As a result, the generating functions ofquadrant walks are now well-understood, provided the allowed steps aresmall, that is $S \subset \{-1, 0,1\}^2$. In particular, having smallsteps is crucial for the definition of a certain group of bi-rationaltransformations of the plane. It has been proved that this group is finite ifand only if the corresponding generating function is D-finite (that is, it satisfies a lineardifferential equation with polynomial coefficients). This group is also thekey to the uniform solution of 19 of the 23 small step models possessing afinite group.In contrast, almost nothing is known for walks with arbitrary steps. In thispaper, we extend the definition of the group, or rather of the associatedorbit, to this general case, and generalize the above uniform solution ofsmall step models. When this approach works, it invariably yields a D-finitegenerating function. We apply it to many quadrant problems, including some infinite families.After developing the general theory, we consider the $13\ 110$ two-dimensionalmodels with steps in $\{-2,-1,0,1\}^2$ having at least one $-2$ coordinate. Weprove that only 240 of them have a finite orbit, and solve 231 of them withour method. The 9 remaining models are the counterparts of the 4 models of thesmall step case that resist the uniform solution method (and which are knownto have an algebraic generating function). We conjecture D-finiteness for their generatingfunctions, but only two of them are likely to be algebraic. We also provenon-D-finiteness for the $12\ 870$ models with an infinite orbit, except for16 of them.

The enumeration of planar maps equipped with an Eulerian orientation has attracted attention in both combinatorics and theoretical physics since at least 2000. The case of 4-valent maps is particularly interesting: these orientations are in bijection with properly 3-coloured quadrangulations, while in physics they correspond to configurations of the ice model. We solve both problems -- namely the enumeration of planar Eulerian orientations and of 4-valent planar Eulerian orientations -- by expressing the associated generating functions as the inverses (for the composition of series) of simple hypergeometric series. Using these expressions, we derive the asymptotic behaviour of the number of planar Eulerian orientations, thus proving earlier predictions of Kostov, Zinn-Justin, Elvey Price and Guttmann. This behaviour, $\mu^n /(n \log n)^2$, prevents the associated generating functions from being D-finite. Still, these generating functions are differentially algebraic, as they satisfy non-linear differential equations of order $2$. Differential algebraicity has recently been proved for other map problems, in particular for maps equipped with a Potts model. Our solutions mix recursive and bijective ingredients. In particular, a preliminary bijection transforms our oriented maps into maps carrying a height function on their vertices. In the 4-valent case, we also observe an unexpected connection with the enumeration of maps equipped with a spanning tree that is internally inactive in the sense of Tutte. This connection remains to be explained combinatorially., Comment: 37 pp

In the 1970s, William Tutte developed a clever algebraic approach, based on certain "invariants", to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past 20 years to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps, taken in $\{-1, 0,1\}^2$. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic. This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity follows almost automatically. Then, we move to a complex analytic viewpoint that has already proved very powerful, leading in particular to integral expressions for the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions for the generating function, and a proof that this series is D-algebraic (that is, satisfies polynomial differential equations)., Comment: 59 pages, 10 figures, 11 tables

We study planar walks that start from a given point (i\_0, j\_0), take their steps in a finite set S, and are confined in the first quadrant of the plane. Their enumeration can be attacked in a systematic way: the generating function Q(x, y, t) that counts them by their length (variable t) and the coordinates of their endpoint (variables x, y) satisfies a linear functional equation encoding the step-by-step description of walks. For instance, for the square lattice walks starting from the origin, this equation reads (xy-t(x + y + x^2 y + x y^2)) Q(x, y, t) = xy-xtQ(x, 0, t)-ytQ(O, y, t). The central question addressed in this paper is the nature of the series Q(x, y , t). When is it algebraic? When is it D-finite (or holonomic)? Can these properties be derived from the functional equation itself? Our first result is a new proof of an old theorem due to Kreweras, according to which one of these walk models has, for mysterious reasons, an algebraic generating function. Then, we provide a new proof of a holonomy criterion recently proved by M. Petkovsek and the author. In both cases, we work directly from the functional equation.

An element w of a Coxeter group W is said to be fully commutative, if any reduced expression of w can be obtained from any other by transposing adjacent pairs of generators. These elements were described in 1996 by Stembridge in the case of finite irreducible groups, and more recently by Biagioli, Jouhet and Nadeau (BJN) in the affine cases. We focus here on the length enumeration of these elements. Using a recursive description, BJN established for the associated generating functions systems of non-linear q-equations. Here, we show that an alternative recursive description leads to explicit expressions for these generating functions., Comment: 37 pages, 7 figures

The number of planar Eulerian maps with n edges is well-known to have a simple expression. But what is the number of planar Eulerian orientations with n edges? This problem appears to be difficult. To approach it, we define and count families of subsets and supersets of planar Eulerian orientations, indexed by an integer k, that converge to the set of all planar Eulerian orientations as k increases. The generating functions of our subsets can be characterized by systems of polynomial equations, and are thus algebraic. The generating functions of our supersets are characterized by polynomial systems involving divided differences, as often occurs in map enumeration. We prove that these series are algebraic as well. We obtain in this way lower and upper bounds on the growth rate of planar Eulerian orientations, which appears to be around 12.5.

In the 1970s, Tutte developed a clever algebraic approach, based on certain "invariants" , to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past decade to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic (with one small exception). This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity comes out (almost) automatically. Then, we move to an analytic viewpoint which has already proved very powerful, leading in particular to integral expressions of the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions of the generating function, and a proof that this series is differentially algebraic (that is, satisfies a non-linear differential equation)., Comment: Extended abstract presented at the conference FPSAC 2016, Vancouver. FPSAC 2016, Jul 2016, Vancouver, Canada. FPSAC 2016 (Formal power series and algebraic combinatorics)

In the past decade, a lot of attention has been devoted to the enumera-tion of walks with prescribed steps confined to a convex cone. In two dimensions, this means counting walks in the first quadrant of the plane (possibly after a linear transformation). But what about walks in non-convex cones? We investigate the two most natural cases: first, square lattice walks avoiding the negative quadrant Q 1 = {(i, j) : i \textless{} 0 and j \textless{} 0}, and then, square lattice walks avoiding the West quadrant Q 2 = {(i, j) : i \textless{} j and i \textless{} --j}. In both cases, the generating function that counts walks starting from the origin is found to differ from a simple D-finite series by an algebraic one. We also obtain closed form expressions for the number of n-step walks ending at certain prescribed endpoints, as a sum of three hypergeometric terms. One of these terms already appears in the enumeration of square lattice walks confined to the cone {(i, j) : i +j $\ge$ 0 and j $\ge$ 0}, known as Gessel's walks. In fact, the enumeration of Gessel's walks follows, by the reflection principle, from the enumeration of walks starting from (--1, 0) and avoiding Q 1. Their generating function turns out to be purely algebraic (as the generating function of Gessel's walks). Another approach to Gessel's walks consists in counting walks that start from (--1, 1) and avoid the West quadrant Q 2. The associated generating function is D-finite but transcendental., Comment: 32 p in Journal of Combinatorial Theory, Series A, Elsevier, 2016, Special issue for the 50th anniversary of the journal

We address the enumeration of q-coloured planar maps counted bythe number of edges and the number of monochromatic edges. We prove that the associated generating function is differentially algebraic,that is, satisfies a non-trivial polynomial differential equation withrespect to the edge variable. We give explicitly a differential systemthat characterizes this series. We then prove a similar result for planar triangulations, thus generalizing a result of Tutte dealing with their proper q-colourings. Instatistical physics terms, we solvethe q-state Potts model on random planar lattices. This work follows a first paper by the same authors, where the generating functionwas proved to be algebraic for certain values of q,including q=1, 2 and 3. It isknown to be transcendental in general. In contrast, our differential system holds for an indeterminate q.For certain special cases of combinatorial interest (four colours; properq-colourings; maps equipped with a spanning forest), we derive from this system, in the case of triangulations, an explicit differential equation of order 2 defining the generating function. For general planar maps, we also obtain a differential equation of order 3 for the four-colour case and for the self-dual Potts model., Comment: 43 pp

Around 2000, Ira Gessel conjectured that the number of lattice walks in the quadrant N^2, starting and ending at the origin (0,0) and taking their steps in {E,NE,W,SW} had a simple hypergeometric form. In the following decade, this problem was recast in the systematic study of walks with small steps (that is,steps in {-1,0,1}^2) confined to the quadrant. The generating functions of such walks are archetypal solutions of partial discrete differential equations.A complete classification of quadrant walks according to the nature of their generating function(algebraic, D-finite or not) is now available, but Gessel'swalks remained mysterious because they were the only model among the 23D-finite ones that had not been given an elementarysolution. Instead, Gessel's conjecture was first proved usingan inventive computer algebra approach in 2008. A year later, the associated three-variate generating function was proved to be algebraic by a computer algebra tour de force. This was re-proved recently using elaborate complex analysis machinery. We give here an elementary and constructive proof. Our approach also solves other quadrant models (with multiple steps) recently proved to be algebraic via computer algebra., Comment: in Advances in Mathematics, Elsevier, 2017

Many recent papers deal with the enumeration of 2-dimensional walks with prescribed steps confined to the positive quadrant. The classification is now complete for walks with steps in $\{0, \pm 1\}^2$: the generating function is D-finite if and only if a certain group associated with the step set is finite. We explore in this paper the analogous problem for 3-dimensional walks confined to the positive octant. The first difficulty is their number: there are 11074225 non-trivial and non-equivalent step sets in $\{0, \pm 1\}^3$ (instead of 79 in the quadrant case). We focus on the 35548 that have at most six steps. We apply to them a combined approach, first experimental and then rigorous. On the experimental side, we try to guess differential equations. We also try to determine if the associated group is finite. The largest finite groups that we find have order 48 -- the larger ones have order at least 200 and we believe them to be infinite. No differential equation has been detected in those cases. On the rigorous side, we apply three main techniques to prove D-finiteness. The algebraic kernel method, applied earlier to quadrant walks, works in many cases. Certain, more challenging, cases turn out to have a special Hadamard structure, which allows us to solve them via a reduction to problems of smaller dimension. Finally, for two special cases, we had to resort to computer algebra proofs. We prove with these techniques all the guessed differential equations. This leaves us with exactly 19 very intriguing step sets for which the group is finite, but the nature of the generating function still unclear., Comment: Final version, to appear in Annals of Combinatorics. 36 pages

At the end of the 1960s, Knuth characterised the permutations that can be sorted using a stack in terms of forbidden patterns. He also showed that they are in bijection with Dyck paths and thus counted by the Catalan numbers. Subsequently, Even \& Itai, Pratt and Tarjan studied permutations that can be sorted using two stacks in parallel. This problem is significantly harder. In particular, a sortable permutation can now be sorted by several distinct sequences of stack operations. Moreover, in order to be sortable, a permutation must avoid infinitely many patterns. The associated counting question has remained open for 40 years. We solve it by giving a pair of functional equations that characterise the generating function of permutations that can be sorted with two parallel stacks. The first component of this system describes the generating function Q(a,u) of square lattice loops confined to the positive quadrant, counted by the length and the number of North-West and East-South factors. Our analysis of the asymptotic number of sortable permutations relies at the moment on two intriguing conjectures dealing with the series Q(a,u). We prove that they hold for loops confined to the upper half plane, or not confined at all. They remain open for quarter plane loops. Given the recent activity on walks confined to cones, we believe them to be attractive per se., Comment: 35 pp

Unranked trees can be represented using their minimal dag (directed acyclic graph). For XML this achieves high compression ratios due to their repetitive mark up. Unranked trees are often represented through first child/next sibling (fcns) encoded binary trees. We study the difference in size (= number of edges) of minimal dag versus minimal dag of the fcns encoded binary tree. One main finding is that the size of the dag of the binary tree can never be smaller than the square root of the size of the minimal dag, and that there are examples that match this bound. We introduce a new combined structure, the hybrid dag, which is guaranteed to be smaller than (or equal in size to) both dags. Interestingly, we find through experiments that last child/previous sibling encodings are much better for XML compression via dags, than fcns encodings. We determine the average sizes of unranked and binary dags over a given set of labels (under uniform distribution) in terms of their exact generating functions, and in terms of their asymptotical behavior., Comment: A short version of this paper appeared in the Proceedings of ICDT 2013

We address the enumeration of p-valent planar maps equipped with a spanning forest, with a weight z per face and a weight u per connected component of the forest. Equivalently, we count p-valent maps equipped with a spanning tree, with a weight z per face and a weight \mu:=u+1 per internally active edge, in the sense of Tutte; or the (dual) p-angulations equipped with a recurrent sandpile configuration, with a weight z per vertex and a variable \mu:=u+1 that keeps track of the level of the configuration. This enumeration problem also corresponds to the limit q -> 0 of the q-state Potts model on p-angulations. Our approach is purely combinatorial. The associated generating function, denoted F(z,u), is expressed in terms of a pair of series defined implicitly by a system involving doubly hypergeometric series. We derive from this system that F(z,u) is differentially algebraic in z, that is, satisfies a differential equation in z with polynomial coefficients in z and u. This has recently been proved to hold for the more general Potts model on 3-valent maps, but via a much more involved and less combinatorial proof. For u >= -1, we study the singularities of F(z,u) and the corresponding asymptotic behaviour of its n-th coefficient. For u>0, we find the standard asymptotic behaviour of planar maps, with a subexponential term in n^{-5/2}. At u=0 we witness a phase transition with a term n^{-3}. When u\in[-1,0), we obtain an extremely unusual behaviour in n^{-3}(\ln n)^{-2}. To our knowledge, this is a new "universality class" for planar maps., Comment: 41 pp

Let A be a minor-closed class of labelled graphs, and let G_n be a random graph sampled uniformly from the set of n-vertex graphs of A. When n is large, what is the probability that G_n is connected? How many components does it have? How large is its biggest component? Thanks to the work of McDiarmid and his collaborators, these questions are now solved when all excluded minors are 2-connected. Using exact enumeration, we study a collection of classes A excluding non-2-connected minors, and show that their asymptotic behaviour may be rather different from the 2-connected case. This behaviour largely depends on the nature of dominant singularity of the generating function C(z) that counts connected graphs of A. We classify our examples accordingly, thus taking a first step towards a classification of minor-closed classes of graphs. Furthermore, we investigate a parameter that has not received any attention in this context yet: the size of the root component. It follows non-gaussian limit laws (beta and gamma), and clearly deserves a systematic investigation., Comment: 37 pages

Consider a rooted binary tree with n nodes. Assign with the root the abscissa 0, and with the left (resp. right) child of a node of abscissa i the abscissa i-1 (resp. i+1). We prove that the number of binary trees of size n having exactly n_i nodes at abscissa i, for l =< i =< r (with n = sum_i n_i), is $$ \frac{n_0}{n_l n_r} {{n_{-1}+n_1} \choose {n_0-1}} \prod_{l\le i\le r \atop i\not = 0}{{n_{i-1}+n_{i+1}-1} \choose {n_i-1}}, $$ with n_{l-1}=n_{r+1}=0. The sequence (n_l, ..., n_{-1};n_0, ..., n_r) is called the vertical profile of the tree. The vertical profile of a uniform random tree of size n is known to converge, in a certain sense and after normalization, to a random mesure called the integrated superbrownian excursion, which motivates our interest in the profile. We prove similar looking formulas for other families of trees whose nodes are embedded in Z. We also refine these formulas by taking into account the number of nodes at abscissa j whose parent lies at abscissa i, and/or the number of vertices at abscissa i having a prescribed number of children at abscissa j, for all i and j. Our proofs are bijective., Comment: 47 pages

An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my}. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T_n^{m}, which generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was introduced by F. Bergeron in connection with the study of diagonal coinvariant spaces in three sets of n variables. The representation of the symmetric group S_n on these spaces is conjectured to be closely related to the natural representation of S_n on (labelled) intervals of the m-Tamari lattice, which we study in this paper. An interval [P,Q] of T_n^{m} is labelled if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. The symmetric group S_n acts on labelled intervals of T_n^{m} by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of S_n. In particular, the dimension of the representation, that is, the number of labelled m-Tamari intervals of size n, is found to be (m+1)^n(mn+1)^{n-2}. These results are new, even when m=1. The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of m-Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. This equation involves two additional variables x and y, a derivative with respect to y and iterated divided differences with respect to x. The hardest part of the proof consists in solving it, and we develop original techniques to do so, partly inspired by previous work on polynomial equations with "catalytic" variables., Comment: 29 pages --- This paper subsumes the research report arXiv:1109.2398, which will not be submitted to any journal

An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T_n^(m), which generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was introduced by F. Bergeron in connection with the study of coinvariant spaces. He conjectured several intriguing formulas dealing with the enumeration of intervals in this lattice. One of them states that the number of intervals in T_n^(m) is $$ \frac {m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}. $$ This conjecture was proved recently, but in a non-bijective way, while its form strongly suggests a connection with plane trees. Here, we prove another conjecture of Bergeron, which deals with the number of labelled, intervals. An interval [P,Q] of T_n^(m) is labelled, if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. We prove that the number of labelled intervals in T_n^(m) is $$ {(m+1)^n(mn+1)^{n-2}}. $$ The form of these numbers suggests a connection with parking functions, but our proof is non-bijective. It is based on a recursive description of intervals, which translates into a functional equation satisfied by the associated generating function. This equation involves a derivative and a divided difference, taken with respect to two additional variables. Solving this equation is the hardest part of the paper. Finding a bijective proof remains an open problem., Comment: 21 pp. This paper is now subsumed by arXiv:1202.5925, and will not be submitted to any journal

In 2010, Duminil-Copin and Smirnov proved a long-standing conjecture of Nienhuis, made in 1982, that the growth constant of self-avoiding walks on the hexagonal (a.k.a. honeycomb) lattice is $\mu=\sqrt{2+\sqrt{2}}.$ A key identity used in that proof was later generalised by Smirnov so as to apply to a general O(n) loop model with $n\in [-2,2]$ (the case $n=0$ corresponding to SAWs). We modify this model by restricting to a half-plane and introducing a surface fugacity $y$ associated with boundary sites (also called surface sites), and obtain a generalisation of Smirnov's identity. The critical value of the surface fugacity was conjectured by Batchelor and Yung in 1995 to be $y_{\rm c}=1+2/\sqrt{2-n}.$ This value plays a crucial role in our generalized identity, just as the value of growth constant did in Smirnov's identity. For the case $n=0$, corresponding to \saws\ interacting with a surface, we prove the conjectured value of the critical surface fugacity. A crucial part of the proof involves demonstrating that the generating function of self-avoiding bridges of height $T$, taken at its critical point $1/\mu$, tends to 0 as $T$ increases, as predicted from SLE theory., Comment: Major revision, references updated, 25 pages, 13 figures

In a 1976 paper published in Science, Knuth presented an algorithm to sample (non-uniform) self-avoiding walks crossing a square of side k. From this sample, he constructed an estimator for the number of such walks. The quality of this estimator is directly related to the (relative) variance of a certain random variable X_k. From his experiments, Knuth suspected that this variance was extremely large (so that the estimator would not be very efficient). But how large? For the analogous Rosenbluth algorithm, which samples unconfined self-avoiding walks of length n, the variance of the corresponding estimator is believed to be exponential in n. A few years ago, Bassetti and Diaconis showed that, for a sampler \`a la Knuth, that generates walks crossing a k\times k square and consisting of North and East steps, the relative variance is only O(\sqrt k). In this note we take one step further and show that, for walks consisting of North, South and East steps, the relative variance jumps to 2^{k(k+1)}/(k+1)^{2k}. This is quasi-exponential in the average length of the walks, which is of order k^2. We also obtain partial results for general self-avoiding walks crossing a square, suggesting that the relative variance could be exponential in k^2 (which is again the average length of these walks). Knuth's algorithm is a basic example of a widely used technique called sequential importance sampling. The present paper, following Bassetti and Diaconis' paper, is one of very few examples where the variance of the estimator can be found., Comment: 19 pages

An m-ballot path of size n is a path on the square grid consisting of north and east steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my}. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice, which generalizes the usual Tamari lattice obtained when m=1. We prove that the number of intervals in this lattice is $$ \frac {m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}. $$ This formula was recently conjectured by Bergeron in connection with the study of coinvariant spaces. The case m=1 was proved a few years ago by Chapoton. Our proof is based on a recursive description of intervals, which translates into a functional equation satisfied by the associated generating function. The solution of this equation is an algebraic series, obtained by a guess-and-check approach. Finding a bijective proof remains an open problem., Comment: 19 pages