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We construct growth bijections for bipolar oriented planar maps and for Schnyder woods. These give direct combinatorial proofs of several counting identities for these objects. Our method mainly uses two ingredients. First, a slit-slide-sew operation, which consists in slightly sliding a map along a well-chosen path. Second, the study of the orbits of natural rerooting operations on the considered classes of oriented maps.

We introduce a model of tree-rooted planar maps weighted by their number of $2$-connected blocks. We study its enumerative properties and prove that it undergoes a phase transition. We give the distribution of the size of the largest $2$-connected blocks in the three regimes (subcritical, critical and supercritical) and further establish that the scaling limit is the Brownian Continuum Random Tree in the critical and supercritical regimes, with respective rescalings $\sqrt{n/\log(n)}$ and $\sqrt{n}$., Comment: 14 pages

We present two graph drawing algorithms based on the recently defined "grand-Schnyder woods", which are a far-reaching generalization of the classical Schnyder woods. The first is a straight-line drawing algorithm for plane graphs with faces of degree 3 and 4 with no separating 3-cycle, while the second is a rectangular drawing algorithm for the dual of such plane graphs. In our algorithms, the coordinates of the vertices are defined in a global manner, based on the underlying grand-Schnyder woods. The grand-Schnyder woods and drawings are computed in linear time. When specializing our algorithms to special classes of plane graphs, we recover the following known algorithms: (1) He's algorithm for rectangular drawing of 3-valent plane graphs, based on transversal structures, (2) Fusy's algorithm for the straight-line drawing of triangulations of the square, based on transversal structures, (3) Bernardi and Fusy's algorithm for the orthogonal drawing of 4-valent plane graphs, based on 2-orientations, (4) Barriere and Huemer's algorithm for the straight-line drawing of quadrangulations, based on separating decompositions. Our contributions therefore provide a unifying perspective on a large family of graph drawing algorithms that were originally defined on different classes of plane graphs and were based on seemingly different combinatorial structures.

A rectangulation is a decomposition of a rectangle into finitely many rectangles. Via natural equivalence relations, rectangulations can be seen as combinatorial objects with a rich structure, with links to lattice congruences, flip graphs, polytopes, lattice paths, Hopf algebras, etc. In this paper, we first revisit the structure of the respective equivalence classes: weak rectangulations that preserve rectangle-segment adjacencies, and strong rectangulations that preserve rectangle-rectangle adjacencies. We thoroughly investigate posets defined by adjacency in rectangulations of both kinds, and unify and simplify known bijections between rectangulations and permutation classes. This yields a uniform treatment of mappings between permutations and rectangulations that unifies the results from earlier contributions, and emphasizes parallelism and differences between the weak and the strong cases. Then, we consider the special case of guillotine rectangulations, and prove that they can be characterized - under all known mappings between permutations and rectangulations - by avoidance of two mesh patterns that correspond to "windmills" in rectangulations. This yields new permutation classes in bijection with weak guillotine rectangulations, and the first known permutation class in bijection with strong guillotine rectangulations. Finally, we address enumerative issues and prove asymptotic bounds for several families of strong rectangulations., Comment: 43 pages, 31 figures

We introduce a simple bijection between Tamari intervals and the blossoming trees (Poulalhon and Schaeffer, 2006) encoding planar triangulations, using a new meandering representation of such trees. Its specializations to the families of synchronized, Kreweras, new/modern, and infinitely modern intervals give a combinatorial proof of the counting formula for each family. Compared to (Bernardi and Bonichon, 2009), our bijection behaves well with the duality of Tamari intervals, enabling also the counting of self-dual intervals., Comment: 37 pages

We define some Schnyder-type combinatorial structures on a class of planar triangulations of the pentagon which are closely related to 5-connected triangulations. The combinatorial structures have three incarnations defined in terms of orientations, corner-labelings, and woods respectively. The wood incarnation consists in 5 spanning trees crossing each other in an orderly fashion. Similarly as for Schnyder woods on triangulations, it induces, for each vertex, a partition of the inner triangles into face-connected regions (5~regions here). We show that the induced barycentric vertex-placement, where each vertex is at the barycenter of the 5 outer vertices with weights given by the number of faces in each region, yields a planar straight-line drawing., Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023)

We define a far-reaching generalization of Schnyder woods which encompasses many classical combinatorial structures on planar graphs. Schnyder woods are defined for planar triangulations as certain triples of spanning trees covering the triangulation and crossing each other in an orderly fashion. They are of theoretical and practical importance, as they are central to the proof that the order dimension of any planar graph is at most 3, and they are also underlying an elegant drawing algorithm. In this article we extend the concept of Schnyder wood well beyond its original setting: for any integer d>2 we define a ``grand-Schnyder'' structure for (embedded) planar graphs which have faces of degree at most d and non-facial cycles of length at least d. We prove the existence of grand-Schnyder structures, provide a linear construction algorithm, describe 4 different incarnations (in terms of tuples of trees, corner labelings, weighted orientations, and marked orientations), and define a lattice for the set of grand Schnyder structures of a given planar graph. We show that the grand-Schnyder framework unifies and extends several classical constructions: Schnyder woods and Schnyder decompositions, regular edge-labelings (a.k.a. transversal structures), and Felsner woods.

Conservative Count-Min, an improved version of Count-Min sketch [Cormode, Muthukrishnan 2005], is an online-maintained hashing-based data structure summarizing element frequency information without storing elements themselves. Although several works attempted to analyze the error that can be made by Count-Min, the behavior of this data structure remains poorly understood. In [Fusy, Kucherov 2022], we demonstrated that under the uniform distribution of input elements, the error of conservative Count-Min follows two distinct regimes depending on its load factor. In this work, we provide a series of experimental results providing new insights into the behavior of conservative Count-Min. Our contributions can be seen as twofold. On one hand, we provide a detailed experimental analysis of the behavior of Count-Min sketch in different regimes and under several representative probability distributions of input elements. On the other hand, we demonstrate improvements that can be made by assigning a variable number of hash functions to different elements. This includes, in particular, reduced space of the data structure while still supporting a small error., Comment: short version to appear in SPIRE'23

In this paper, the scaling limit of random connected cubic planar graphs (respectively multigraphs) is shown to be the Brownian sphere. The proof consists in essentially two main steps. First, thanks to the known decomposition of cubic planar graphs into their 3-connected components, the metric structure of a random cubic planar graph is shown to be well approximated by its unique 3-connected component of linear size, with modified distances. Then, Whitney's theorem ensures that a 3-connected cubic planar graph is the dual of a simple triangulation, for which it is known that the scaling limit is the Brownian sphere. Curien and Le Gall have recently developed a framework to study the modification of distances in general triangulations and in their dual. By extending this framework to simple triangulations, it is shown that 3-connected cubic planar graphs with modified distances converge jointly with their dual triangulation to the Brownian sphere., Comment: 58 pages

Count-Min sketch is a hash-based data structure to represent a dynamically changing associative array of counters. Here we analyse the counting version of Count-Min under a stronger update rule known as \textit{conservative update}, assuming the uniform distribution of input keys. We show that the accuracy of conservative update strategy undergoes a phase transition, depending on the number of distinct keys in the input as a fraction of the size of the Count-Min array. We prove that below the threshold, the relative error is asymptotically $o(1)$ (as opposed to the regular Count-Min strategy), whereas above the threshold, the relative error is $\Theta(1)$. The threshold corresponds to the peelability threshold of random $k$-uniform hypergraphs. We demonstrate that even for small number of keys, peelability of the underlying hypergraph is a crucial property to ensure the $o(1)$ error. Finally, we provide an experimental evidence that the phase transition does not extend to non-uniform distributions, in particular to the popular Zipf's distribution., Comment: 19 pages, 4 figures

We show that corner polyhedra and 3-connected Schnyder labelings join the growing list of planar structures that can be set in exact correspondence with (weighted) models of quadrant walks via a bijection due to Kenyon, Miller, Sheffield and Wilson. Our approach leads to a first polynomial time algorithm to count these structures, and to the determination of their exact asymptotic growth constants: the number $p_n$ of corner polyhedra and $s_n$ of 3-connected Schnyder labelings of size $n$ respectively satisfy $(p_n)^{1/n}\to 9/2$ and $(s_n)^{1/n}\to 16/3$ as $n$ goes to infinity. While the growth rates are rational, like in the case of previously known instances of such correspondences, the exponent of the asymptotic polynomial correction to the exponential growth does not appear to follow from the now standard Denisov-Wachtel approach, due to a bimodal behavior of the step set of the underlying tandem walk. However a heuristic argument suggests that these exponents are $-1-\pi/\arccos(9/16)\approx -4.23$ for $p_n$ and $-1-\pi/\arccos(22/27)\approx -6.08$ for $s_n$, which would imply that the associated series are not D-finite., Comment: 28 pages

We show that plane bipolar posets (i.e., plane bipolar orientations with no transitive edge) and transversal structures can be set in correspondence to certain (weighted) models of quadrant walks, via suitable specializations of a bijection due to Kenyon, Miller, Sheffield and Wilson. We then derive exact and asymptotic counting results. In particular we prove (computationally and then bijectively) that the number of plane bipolar posets on $n+2$ vertices equals the number of plane permutations of size $n$. Regarding transversal structures, for each $v\geq 0$ we consider $t_n(v)$ the number of such structures with $n+4$ vertices and weight $v$ per quadrangular inner face (the case $v=0$ corresponds to having only triangular inner faces). We obtain a recurrence to compute $t_n(v)$, and an asymptotic formula that for $v=0$ gives $t_n(0)\sim c\ \!(27/2)^nn^{-1-\pi/\mathrm{arccos}(7/8)}$ for some $c>0$, which also ensures that the associated generating function is not D-finite., Comment: 31 pages

We introduce bijections between families of rooted maps with unfixed genus and families of so-called blossoming trees endowed with an arbitrary forward matching of their leaves. We first focus on Eulerian maps with controlled vertex degrees. The mapping from blossoming trees to maps is a generalization to unfixed genus of Schaeffer's closing construction for planar Eulerian maps. The inverse mapping relies on the existence of canonical orientations which allow to equip the maps with canonical spanning trees, as proved by Bernardi. Our bijection gives in particular (here in the Eulerian case) a combinatorial explanation to the striking similarity between the (infinite) recursive system of equations which determines the partition function of maps with unfixed genus (as obtained via matrix models and orthogonal polynomials) and that determining the partition function of planar maps. All the functions in the recursive system get a combinatorial interpretation as generating functions for maps endowed with particular multiple markings of their edges. This allows us in particular to give a combinatorial proof of some differential identities satisfied by these functions. We also consider face-colored Eulerian maps with unfixed genus and derive some striking identities between their generating functions and those of properly weighted marked maps. The same methodology is then applied to deal with $m$-regular bipartite maps with unfixed genus, leading to similar results. The case of cubic maps is also briefly discussed., Comment: 44 pages, 20 figures

We investigate polyharmonic functions associated to Brownian motion and random walks in cones. These are functions which cancel some power of the usual Laplacian in the continuous setting and of the discrete Laplacian in the discrete setting. We show that polyharmonic functions naturally appear while considering asymptotic expansions of the heat kernel in the Brownian case and in lattice walk enumeration problems. We provide a method to construct general polyharmonic functions through Laplace transforms and generating functions in the continuous and discrete cases, respectively. This is done by using a functional equation approach., Comment: To appear in the proceedings of the 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA2020). 18 pages

We present a bijection for toroidal maps that are essentially $3$-connected ($3$-connected in the periodic planar representation). Our construction actually proceeds on certain closely related bipartite toroidal maps with all faces of degree $4$ except for a hexagonal root-face. We show that these maps are in bijection with certain well-characterized bipartite unicellular maps. Our bijection, closely related to the recent one by Bonichon and L\'ev\^eque for essentially 4-connected toroidal triangulations, can be seen as the toroidal counterpart of the one developed in the planar case by Fusy, Poulalhon and Schaeffer, and it extends the one recently proposed by Fusy and L\'ev\^eque for essentially simple toroidal triangulations. Moreover, we show that rooted essentially $3$-connected toroidal maps can be decomposed into two pieces, a toroidal part that is treated by our bijection, and a planar part that is treated by the above-mentioned planar case bijection. This yields a combinatorial derivation for the bivariate generating function of rooted essentially $3$-connected toroidal maps, counted by vertices and faces., Comment: 30 pages

Generalized Tamari intervals have been recently introduced by Pr\'eville-Ratelle and Viennot, and have been proved to be in bijection with (rooted planar) non-separable maps by Fang and Pr\'eville-Ratelle. We present two new bijections between generalized Tamari intervals and non-separable maps. Our first construction proceeds via separating decompositions on simple bipartite quadrangulations (which are known to be in bijection with non-separable maps). It can be seen as an extension of the Bernardi-Bonichon bijection between Tamari intervals and minimal Schnyder woods. On the other hand, our second construction relies on a specialization of the Bernardi-Bonichon bijection to so-called synchronized Tamari intervals, which are known to be in one-to-one correspondence with generalized Tamari intervals. It yields a trivariate generating function expression that interpolates between the bivariate generating function for generalized Tamari intervals, and the univariate generating function for Tamari intervals., Comment: 22 pages

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 consider the graphs involved in the theoretical physics model known as the colored Sachdev-Ye-Kitaev (SYK) model. We study in detail their combinatorial properties at any order in the so-called $1/N$ expansion, and we enumerate these graphs asymptotically. Because of the duality between colored graphs involving $q+1$ colors and colored triangulations in dimension $q$, our results apply to the asymptotic enumeration of spaces that generalize unicellular maps - in the sense that they are obtained from a single building block - for which a higher-dimensional generalization of the genus is kept fixed., Comment: 22 pages, 13 figures

We present unified bijections for maps on the torus with control on the face-degrees and essential girth (girth of the periodic planar representation). A first step is to show that for d>=3 every toroidal d-angulation of essential girth d can be endowed with a certain "canonical" orientation (formulated as a weight-assignment on the half-edges). Using an adaptation of a construction by Bernardi and Chapuy, we can then derive a bijection between face-rooted toroidal d-angulations of essential girth d (with the condition that, apart from the root-face contour, no other closed walk of length d encloses the root-face) and a family of decorated unicellular maps. The orientations and bijections can then be generalized, for any d>=1, to toroidal face-rooted maps of essential girth d with a root-face of degree d (and with the same root-face contour condition as for d-angulations), and they take a simpler form in the bipartite case, as a parity specialization. On the enumerative side we obtain explicit algebraic expressions for the generating functions of rooted essentially simple triangulations and bipartite quadrangulations on the torus. Our bijective constructions can be considered as toroidal counterparts of those obtained by Bernardi and the first author in the planar case, and they also build on ideas introduced by Despr\'e, Gon\c{c}alves and the second author for essentially simple triangulations, of imposing a balancedness condition on the orientations in genus 1., Comment: 69 pages

In the study of lattice walks there are several examples of enumerative equivalences which amount to a trade-off between domain and endpoint constraints. We present a family of such bijections for simple walks in Weyl chambers which use arc diagrams in a natural way. One consequence is a set of new bijections for standard Young tableaux of bounded height. A modification of the argument in two dimensions yields a bijection between Baxter permutations and walks ending on an axis, answering a recent question of Burrill et al. (2016). Some of our arguments (and related results) are proved using Schnyder woods. Our strategy for simple walks extends to any dimension and yields a new bijective connection between standard Young tableaux of height at most $2k$ and certain walks with prescribed endpoints in the $k$-dimensional Weyl chamber of type D., Comment: This is a full version, published in the European Journal of Combinatorics. It is 19 pages long and have 8 Figures

The Catalan numbers count many classes of combinatorial objects. The most emblematic such objects are probably the Dyck walks and the binary trees, and, whenever another class of combinatorial objects is counted by the Catalan numbers, it is natural to search for an explicit bijection between the latter objects and one of the former objects. In most cases, such a bijection happens to be relatively simple but it might sometimes be more intricate. In this work, we focus on so-called \emph{basketball walks}, which are integer-valued walks with step-set $\{-2,-1,+1,+2\}$. The presence of $-2$ as an allowed step makes it impossible to use the classical {\L}ukasiewicz encoding of trees by integer-valued walks, and thus a different strategy is needed. We give an explicit bijection that maps, for each $n\ge 2$, $n$-step basketball walks from $0$ to $0$ that visit $1$ and are positive except at their extremities to $n$-leaf binary trees. Moreover, we can partition the steps of a walk into $\pm 1$-steps, odd $+2$-steps or even $-2$-steps, and odd $-2$-steps or even $+2$-steps, and these three types of steps are mapped through our bijection to double leaves, left leaves, and right leaves of the corresponding tree. We also prove that basketball walks from $0$ to $1$ that are positive except at the origin are in bijection with increasing unary-binary trees with associated permutation avoiding $213$. We furthermore give the refined generating function of these objects with an extra variable accounting for the unary nodes.

Distance-hereditary graphs form an important class of graphs, from the theoretical point of view, due to the fact that they are the totally decomposable graphs for the split-decomposition. The previous best enumerative result for these graphs is from Nakano et al. (J. Comp. Sci. Tech., 2007), who have proven that the number of distance-hereditary graphs on $n$ vertices is bounded by ${2^{\lceil 3.59n\rceil}}$. In this paper, using classical tools of enumerative combinatorics, we improve on this result by providing an exact enumeration of distance-hereditary graphs, which allows to show that the number of distance-hereditary graphs on $n$ vertices is tightly bounded by ${(7.24975\ldots)^n}$---opening the perspective such graphs could be encoded on $3n$ bits. We also provide the exact enumeration and asymptotics of an important subclass, the 3-leaf power graphs. Our work illustrates the power of revisiting graph decomposition results through the framework of analytic combinatorics., Comment: 14 pages, 4 figures, and a Maple worksheet; preprint of long version of IGCT 2014 article presented as "An Enumeration of Distance-Hereditary and 3-Leaf Power Graphs."

Billey et al. [arXiv:1507.04976] have recently discovered a surprisingly simple formula for the number $a_n(\sigma)$ of leaf-labelled rooted non-embedded binary trees (also known as phylogenetic trees) with $n\geq 1$ leaves, fixed (for the relabelling action) by a given permutation $\sigma\in\frak{S}_n$. Denoting by $\lambda\vdash n$ the integer partition giving the sizes of the cycles of $\sigma$ in non-increasing order, they show by a guessing/checking approach that if $\lambda$ is a binary partition (it is known that $a_n(\sigma)=0$ otherwise), then $$ a_n(\sigma)=\prod_{i=2}^{\ell(\lambda)}(2(\lambda_i+\cdots+\lambda_{\ell(\lambda)})-1), $$ and they derive from it a formula and random generation procedure for tanglegrams (and more generally for tangled chains). Our main result is a combinatorial proof of the formula, which yields a simplification of the random sampler for tangled chains., Comment: 6 pages

We present bijections for planar maps with boundaries. In particular, we obtain bijections for triangulations and quadrangulations of the sphere with boundaries of prescribed lengths. For triangulations we recover the beautiful factorized formula obtained by Krikun using a (technically involved) generating function approach. The analogous formula for quadrangulations is new. We also obtain a far-reaching generalization for other face-degrees. In fact, all the known enumerative formulas for maps with boundaries are proved bijectively in the present article (and several new formulas are obtained). Our method is to show that maps with boundaries can be endowed with certain "canonical" orientations, making them amenable to the master bijection approach we developed in previous articles. As an application of our enumerative formulas, we note that they provide an exact solution of the dimer model on rooted triangulations and quadrangulations., Comment: 42 pages

We use a continued fraction approach to compare two statistical ensembles of quadrangulations with a boundary, both controlled by two parameters. In the first ensemble, the quadrangulations are bicolored and the parameters control their numbers of vertices of both colors. In the second ensemble, the parameters control instead the number of vertices which are local maxima for the distance to a given vertex, and the number of those which are not. Both ensembles may be described either by their (bivariate) generating functions at fixed boundary length or, after some standard slice decomposition, by their (bivariate) slice generating functions. We first show that the fixed boundary length generating functions are in fact equal for the two ensembles. We then show that the slice generating functions, although different for the two ensembles, simply correspond to two different ways of encoding the same quantity as a continued fraction. This property is used to obtain explicit expressions for the slice generating functions in a constructive way., Comment: 34 pages, 17 figures

Walks on Young's lattice of integer partitions encode many objects of algebraic and combinatorial interest. Chen et al. established connections between such walks and arc diagrams. We show that walks that start at $\varnothing$, end at a row shape, and only visit partitions of bounded height are in bijection with a new type of arc diagram -- open diagrams. Remarkably two subclasses of open diagrams are equinumerous with well known objects: standard Young tableaux of bounded height, and Baxter permutations. We give an explicit combinatorial bijection in the former case., Comment: 20 pages; Text overlap with arXiv:1411.6606. This is the full version of that extended abstract. Conjectures from that work are proved in this work

We compute the distance-dependent two-point function of vertex-bicolored planar maps, i.e., maps whose vertices are colored in black and white so that no adjacent vertices have the same color. By distance-dependent two-point function, we mean the generating function of these maps with both a marked oriented edge and a marked vertex which are at a prescribed distance from each other. As customary, the maps are enumerated with arbitrary degree-dependent face weights, but the novelty here is that we also introduce color-dependent vertex weights. Explicit expressions are given for vertex-bicolored maps with bounded face degrees in the form of ratios of determinants of fixed size. Our approach is based on a slice decomposition of maps which relates the distance-dependent two-point function to the coefficients of the continued fraction expansions of some distance-independent map generating functions. Special attention is paid to the case of vertex-bicolored quadrangulations and hexangulations, whose two-point functions are also obtained in a more direct way involving equivalences with hard dimer statistics. A few consequences of our results, as well as some extension to vertex-tricolored maps, are also discussed., Comment: 43 pages, 21 figures

Three-dimensional random tensor models are a natural generalization of the celebrated matrix models. The associated tensor graphs, or 3D maps, can be classified with respect to a particular integer or half-integer, the degree of the respective graph. In this paper we analyze the general term of the asymptotic expansion in N, the size of the tensor, of a particular random tensor model, the multi-orientable tensor model. We perform their enumeration and we establish which are the dominant configurations of a given degree., Comment: 27 pages, 24 figures, several minor modifications have been made, one figure has been added; accepted for publication in "Electronic Journal of Combinatorics"

We present a general bijective approach to planar hypermaps with two main results. First we obtain unified bijections for all classes of maps or hypermaps defined by face-degree constraints and girth constraints. To any such class we associate bijectively a class of plane trees characterized by local constraints. This unifies and greatly generalizes several bijections for maps and hypermaps. Second, we present yet another level of generalization of the bijective approach by considering classes of maps with non-uniform girth constraints. More precisely, we consider "well-charged maps", which are maps with an assignment of "charges" (real numbers) on vertices and faces, with the constraints that the length of any cycle of the map is at least equal to the sum of the charges of the vertices and faces enclosed by the cycle. We obtain a bijection between charged hypermaps and a class of plane trees characterized by local constraints.

We compute the distance-dependent three-point function of general planar maps and of bipartite planar maps, i.e., the generating function of these maps with three marked vertices at prescribed pairwise distances. Explicit expressions are given for maps counted by their number of edges only, or by both their numbers of edges and faces. A few limiting cases and applications are discussed., Comment: 33 pages, 12 figures

We consider the problem of computing the distance-dependent two-point function of general planar maps and hypermaps, i.e. the problem of counting such maps with two marked points at a prescribed distance. The maps considered here may have faces of arbitrarily large degree, which requires new bijections to be tackled. We obtain exact expressions for the following cases: general and bipartite maps counted by their number of edges, 3-hypermaps and 3-constellations counted by their number of dark faces, and finally general and bipartite maps counted by both their number of edges and their number of faces., Comment: 32 pages, 17 figures

We extend the notion of canonical ordering (initially developed for planar triangulations and 3-connected planar maps) to cylindric (essentially simple) triangulations and more generally to cylindric (essentially internally) $3$-connected maps. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case) and of Kant (in the $3$-connected case) to this setting. Precisely, for any cylindric essentially internally $3$-connected map $G$ with $n$ vertices, we can obtain in linear time a periodic (in $x$) straight-line drawing of $G$ that is crossing-free and internally (weakly) convex, on a regular grid $\mathbb{Z}/w\mathbb{Z}\times[0..h]$, with $w\leq 2n$ and $h\leq n(2d+1)$, where $d$ is the face-distance between the two boundaries. This also yields an efficient periodic drawing algorithm for graphs on the torus. Precisely, for any essentially $3$-connected map $G$ on the torus (i.e., $3$-connected in the periodic representation) with $n$ vertices, we can compute in linear time a periodic straight-line drawing of $G$ that is crossing-free and (weakly) convex, on a periodic regular grid $\mathbb{Z}/w\mathbb{Z}\times\mathbb{Z}/h\mathbb{Z}$, with $w\leq 2n$ and $h\leq 1+2n(c+1)$, where $c$ is the face-width of $G$. Since $c\leq\sqrt{2n}$, the grid area is $O(n^{5/2})$., Comment: 37 pages

We obtain a very simple formula for the generating function of bipartite (resp. quasi-bipartite) planar maps with boundaries (holes) of prescribed lengths, which generalizes certain expressions obtained by Eynard in a book to appear. The formula is derived from a bijection due to Bouttier, Di Francesco and Guitter combined with a process (reminiscent of a construction of Pitman) of aggregating connected components of a forest into a single tree. The formula naturally extends to $p$-constellations and quasi-$p$-constellations with boundaries (the case $p=2$ corresponding to bipartite maps)., Comment: 23 pages, full paper version of v1, with results extended to constellations and quasi constellations

We show that the diameter D(G_n) of a random labelled connected planar graph with n vertices is equal to n^{1/4+o(1)}, in probability. More precisely there exists a constant c>0 such that the probability that D(G_n) lies in the interval (n^{1/4-\epsilon},n^{1/4+\epsilon}) is greater than 1-\exp(-n^{c\epsilon}) for {\epsilon} small enough and n>n_0(\epsilon). We prove similar statements for 2-connected and 3-connected planar graphs and maps., Comment: 24 pages, 7 figures

We consider unicellular maps, or polygon gluings, of fixed genus. A few years ago the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of Catalan numbers in counting formulas for these objects. In this paper, we give another bijection that explicitly describes the "recursive part" of the first bijection. As a result we obtain a very simple description of unicellular maps as pairs made by a plane tree and a permutation-like structure. All the previously known formulas follow as an immediate corollary or easy exercise, thus giving a bijective proof for each of them, in a unified way. For some of these formulas, this is the first bijective proof, e.g. the Harer-Zagier recurrence formula, the Lehman-Walsh formula and the Goupil-Schaeffer formula. We also discuss several applications of our construction: we obtain a new proof of an identity related to covered maps due to Bernardi and the first author, and thanks to previous work of the second author, we give a new expression for Stanley character polynomials, which evaluate irreducible characters of the symmetric group. Finally, we show that our techniques apply partially to unicellular 3-constellations and to related objects that we call quasi-constellations., Comment: v5: minor revision after reviewers comments, 33 pages, added a refinement by degree of the Harer-Zagier formula and more details in some proofs

It is a classical result of Stein and Waterman that the asymptotic number of RNA secondary structures is $1.104366 \cdot n^{-3/2} \cdot 2.618034^n$. Motivated by the kinetics of RNA secondary structure formation, we are interested in determining the asymptotic number of secondary structures that are locally optimal, with respect to a particular energy model. In the Nussinov energy model, where each base pair contributes -1 towards the energy of the structure, locally optimal structures are exactly the saturated structures, for which we have previously shown that asymptotically, there are $1.07427\cdot n^{-3/2} \cdot 2.35467^n$ many saturated structures for a sequence of length $n$. In this paper, we consider the base stacking energy model, a mild variant of the Nussinov model, where each stacked base pair contributes -1 toward the energy of the structure. Locally optimal structures with respect to the base stacking energy model are exactly those secondary structures, whose stems cannot be extended. Such structures were first considered by Evers and Giegerich, who described a dynamic programming algorithm to enumerate all locally optimal structures. In this paper, we apply methods from enumerative combinatorics to compute the asymptotic number of such structures. Additionally, we consider analogous combinatorial problems for secondary structures with annotated single-stranded, stacking nucleotides (dangles)., Comment: 27 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

This article presents new enumerative results related to symmetric planar maps. In the first part a new way of enumerating rooted simple quadrangulations and rooted simple triangulations is presented, based on the description of two different quotient operations on symmetric simple quadrangulations and triangulations. In the second part, based on results of Bouttier, Di Francesco and Guitter and on quotient and substitution operations, the series of three families of symmetric quadrangular and triangular dissections of polygons are computed, with control on the distance from the central vertex to the outer boundary., Comment: 20 pages, long version of proceedings at Eurocomb 2011. Supported by the European project ExploreMaps ERC StG 208471

This article presents unified bijective constructions for planar maps, with control on the face degrees and on the girth. Recall that the girth is the length of the smallest cycle, so that maps of girth at least $d=1,2,3$ are respectively the general, loopless, and simple maps. For each positive integer $d$, we obtain a bijection for the class of plane maps (maps with one distinguished root-face) of girth $d$ having a root-face of degree $d$. We then obtain more general bijective constructions for annular maps (maps with two distinguished root-faces) of girth at least $d$. Our bijections associate to each map a decorated plane tree, and non-root faces of degree $k$ of the map correspond to vertices of degree $k$ of the tree. As special cases we recover several known bijections for bipartite maps, loopless triangulations, simple triangulations, simple quadrangulations, etc. Our work unifies and greatly extends these bijective constructions. In terms of counting, we obtain for each integer $d$ an expression for the generating function $F_d(x_d,x_{d+1},x_{d+2},...)$ of plane maps of girth $d$ with root-face of degree $d$, where the variable $x_k$ counts the non-root faces of degree $k$. The expression for $F_1$ was already obtained bijectively by Bouttier, Di Francesco and Guitter, but for $d\geq 2$ the expression of $F_d$ is new. We also obtain an expression for the generating function $\G_{p,q}^{(d,e)}(x_d,x_{d+1},...)$ of annular maps with root-faces of degrees $p$ and $q$, such that cycles separating the two root-faces have length at least $e$ while other cycles have length at least $d$. Our strategy is to obtain all the bijections as specializations of a single "master bijection" introduced by the authors in a previous article. In order to use this approach, we exhibit certain "canonical orientations" characterizing maps with prescribed girth constraints.

We enumerate bijectively the family of involutive Baxter permutations according to various parameters; in particular we obtain an elementary proof that the number of involutive Baxter permutations of size $2n$ with no fixed points is $\frac{3\cdot 2^{n-1}}{(n+1)(n+2)}\binom{2n}{n}$, a formula originally discovered by M. Bousquet-M\'elou using generating functions. The same coefficient also enumerates planar maps with $n$ edges, endowed with an acyclic orientation having a unique source, and such that the source and sinks are all incident to the outer face., Comment: 8 pages

Schnyder woods are decompositions of simple triangulations into three edge-disjoint spanning trees crossing each other in a specific way. In this article, we define a generalization of Schnyder woods to $d$-angulations (plane graphs with faces of degree $d$) for all $d\geq 3$. A \emph{Schnyder decomposition} is a set of $d$ spanning forests crossing each other in a specific way, and such that each internal edge is part of exactly $d-2$ of the spanning forests. We show that a Schnyder decomposition exists if and only if the girth of the $d$-angulation is $d$. As in the case of Schnyder woods ($d=3$), there are alternative formulations in terms of orientations ("fractional" orientations when $d\geq 5$) and in terms of corner-labellings. Moreover, the set of Schnyder decompositions on a fixed $d$-angulation of girth $d$ is a distributive lattice. We also show that the structures dual to Schnyder decompositions (on $d$-regular plane graphs of mincut $d$ rooted at a vertex $v^*$) are decompositions into $d$ spanning trees rooted at $v^*$ such that each edge not incident to $v^*$ is used in opposite directions by two trees. Additionally, for even values of $d$, we show that a subclass of Schnyder decompositions, which are called even, enjoy additional properties that yield a reduced formulation; in the case d=4, these correspond to well-studied structures on simple quadrangulations (2-orientations and partitions into 2 spanning trees). In the case d=4, the dual of even Schnyder decompositions yields (planar) orthogonal and straight-line drawing algorithms. For a 4-regular plane graph $G$ of mincut 4 with $n$ vertices plus a marked vertex $v$, the vertices of $G\backslash v$ are placed on a $(n-1) \times (n-1)$ grid according to a permutation pattern, and in the orthogonal drawing each of the $2n-2$ edges of $G\backslash v$ has exactly one bend. Embedding also the marked vertex $v$ is doable at the cost of two additional rows and columns and 8 additional bends for the 4 edges incident to $v$. We propose a further compaction step for the drawing algorithm and show that the obtained grid-size is strongly concentrated around $25n/32\times 25n/32$ for a uniformly random instance with $n$ vertices.

A $d$-angulation is a planar map with faces of degree $d$. We present for each integer $d\geq 3$ a bijection between the class of $d$-angulations of girth $d$ (i.e., with no cycle of length less than $d$) and a class of decorated plane trees. Each of the bijections is obtained by specializing a "master bijection" which extends an earlier construction of the first author. Our construction unifies known bijections by Fusy, Poulalhon and Schaeffer for triangulations ($d=3$) and by Schaeffer for quadrangulations ($d=4$). For $d\geq 5$, both the bijections and the enumerative results are new. We also extend our bijections so as to enumerate \emph{$p$-gonal $d$-angulations} ($d$-angulations with a simple boundary of length $p$) of girth $d$. We thereby recover bijectively the results of Brown for simple $p$-gonal triangulations and simple $2p$-gonal quadrangulations and establish new results for $d\geq 5$. A key ingredient in our proofs is a class of orientations characterizing $d$-angulations of girth $d$. Earlier results by Schnyder and by De Fraysseix and Ossona de Mendez showed that simple triangulations and simple quadrangulations are characterized by the existence of orientations having respectively indegree 3 and 2 at each inner vertex. We extend this characterization by showing that a $d$-angulation has girth $d$ if and only if the graph obtained by duplicating each edge $d-2$ times admits an orientation having indegree $d$ at each inner vertex.

We present a unified general method for the asymptotic study of graphs from the so-called "subcritical"$ $ graph classes, which include the classes of cacti graphs, outerplanar graphs, and series-parallel graphs. This general method works both in the labelled and unlabelled framework. The main results concern the asymptotic enumeration and the limit laws of properties of random graphs chosen from subcritical classes. We show that the number $g_n/n!$ (resp. $g_n$) of labelled (resp. unlabelled) graphs on $n$ vertices from a subcritical graph class ${G}=\cup_n {G_n}$ satisfies asymptotically the universal behaviour $$ g_n = c n^{-5/2} \gamma^n (1+o(1)) $$ for computable constants $c,\gamma$, e.g. $\gamma\approx 9.38527$ for unlabelled series-parallel graphs, and that the number of vertices of degree $k$ ($k$ fixed) in a graph chosen uniformly at random from $G_n$, converges (after rescaling) to a normal law as $n\to\infty$.

We introduce a general method to count unlabeled combinatorial structures and to efficiently generate them at random. The approach is based on pointing unlabeled structures in an "unbiased" way that a structure of size n gives rise to n pointed structures. We extend Polya theory to the corresponding pointing operator, and present a random sampling framework based on both the principles of Boltzmann sampling and on P\'olya operators. All previously known unlabeled construction principles for Boltzmann samplers are special cases of our new results. Our method is illustrated on several examples: in each case, we provide enumerative results and efficient random samplers. The approach applies to unlabeled families of plane and nonplane unrooted trees, and tree-like structures in general, but also to families of graphs (such as cacti graphs and outerplanar graphs) and families of planar maps., Comment: 41 pages. This is an extended and revised journal version of a conference paper with the title "An unbiased pointing operator for unlabeled structures, with applications to counting and sampling", Proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA'07), 2007, New Orleans. The second Arxiv version incorporates many improvements suggested by anonymous referees of the journal version

It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface S_g of genus g grows asymptotically like $c^{(g)}n^{5(g-1)/2-1}\gamma^n n!$ where $c^{(g)}>0$, and $\gamma \approx 27.23$ is the exponential growth rate of planar graphs. This generalizes the result for the planar case g=0, obtained by Gimenez and Noy. An analogous result for non-orientable surfaces is obtained. In addition, it is proved that several parameters of interest behave asymptotically as in the planar case. It follows, in particular, that a random graph embeddable in S_g has a unique 2-connected component of linear size with high probability.

Schnyder woods are a well-known combinatorial structure for plane triangulations, which yields a decomposition into 3 spanning trees. We extend here definitions and algorithms for Schnyder woods to closed orientable surfaces of arbitrary genus. In particular, we describe a method to traverse a triangulation of genus $g$ and compute a so-called $g$-Schnyder wood on the way. As an application, we give a procedure to encode a triangulation of genus $g$ and $n$ vertices in $4n+O(g \log(n))$ bits. This matches the worst-case encoding rate of Edgebreaker in positive genus. All the algorithms presented here have execution time $O((n+g)g)$, hence are linear when the genus is fixed., Comment: 27 pages, to appear in a special issue of Discrete and Computational Geometry