Generating functions of bipartite maps on orientable surfaces (extended abstract)

International audienceWe compute, for each genus $g$ ≥ 0, the generating function $L$$g$ ≡ $L$$g$($t$;$p$1,$p$2,...) of (labelled) bipartite maps on the orientable surface of genus $g$, with control on all face degrees. We exhibit an explicit change of variables such that for each $g$, $L$$g$ is a rational function in the new variables, computable by an explicit recursion on the genus. The same holds for the generating function $L$$g$ of rooted bipartite maps. The form of the result is strikingly similar to the Goulden/Jackson/Vakil and Goulden/Guay-Paquet/Novak formulas for the generating functions of classical and monotone Hurwitz numbers respectively, which suggests stronger links between these models. Our result strengthens recent results of Kazarian and Zograf, who studied the case where the number of faces is bounded, in the equivalent formalism of dessins d’enfants. Our proofs borrow some ideas from Eynard’s “topological recursion” that he applied in particular to even-faced maps (unconventionally called “bipartite maps” in his work). However, the present paper requires no previous knowledge of this topic and comes with elementary (complex-analysis-free) proofs written in the perspective of formal power series.Nous calculons, pour chaque genre $g$ ≥ 0, la série génératrice $L$$g$ ≡ $L$$g$($t$;$p$1,$p$2,...) des cartes bipartites (étiquetées) sur la surface orientable de genre $g$, avec contrôle des degrés des faces. On exhibe un changement de variable explicite tel que pour tout $g$, $L$$g$ est une fonction rationnelle des nouvelles variables, calculable par une récurrence explicite sur le genre. La même chose est vraie de la série génératrice $L$$g$ des cartes biparties enracinées. La forme du résultat est similaire aux formules de Goulden/Jackson/Vakil et Goulden/Guay-Paquet/Novak pour les séries génératrices de nombres de Hurwitz classiques et monotones, respectivement, ce qui suggère des liens plus forts entre ces modèles. Notre résultat renforce des résultats récents de Kazarian et Zograf, qui étudient le cas où le nombre de faces est borné, dans le formalisme équivalent des dessins d’enfants. Nos démonstrations utilisent deux idées de la “récurrence topologique” d’Eynard, qu’il a appliquée notamment aux cartes paires (appelées de manière non-standard “cartes biparties” dans son travail). Cela dit, ce papier ne requiert pas de connaissance préliminaire sur ce sujet, et nos démonstrations (sans analyse complexe) sont écrites dans le language des séries formelles

A bijection for rooted maps on general surfaces (extended abstract)

International audienceWe extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, orientable or non-orientable. This general construction requires new ideas and is more delicate than the special orientable case, but carries the same information. It thus gives a uniform combinatorial interpretation of the counting exponent $\frac{5(h-1)}{2}$ for both orientable and non-orientable maps of Euler characteristic $2-2h$ and of the algebraicity of their generating functions. It also shows the universality of the renormalization factor $n$¼ for the metric of maps, on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation of size $n$ on any fixed surface converge in distribution. Finally, it also opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold.Nous étendons la bijection de Marcus et Schaeffer entre quadrangulations biparties orientables (de manière équivalente: cartes enracinées) et cartes à une face étiquetées orientables à toutes les surfaces, orientables ou non. Cette construction générale requiert des idées nouvelles et est plus délicate que dans le cas particulier orientable, mais permet des utilisations similaires. Elle donne donc une interprétation combinatoire uniforme de l’exposant de comptage $\frac{5(h-1)}{2}$ pour les cartes orientables et non-orientables de caractéristique d’Euler $2-2h$, et de l’algébricité des fonctions génératrices. Elle montre l’universalité du facteur de normalisation $n$¼ pour la métrique des cartes, sur toutes les surfaces: le profil et le rayon d’une quadrangulation enracinée pointée sur une surface fixée converge en distribution. Enfin, elle ouvre à la voie à l’étude des surfaces Browniennes pour toute 2-variété compacte

Simple recurrence formulas to count maps on orientable surfaces

We establish a simple recurrence formula for the number $Q_g^n$ of rooted orientable maps counted by edges and genus. We also give a weighted variant for the generating polynomial $Q_g^n(x)$ where $x$ is a parameter taking the number of faces of the map into account, or equivalently a simple recurrence formula for the refined numbers $M_g^{i,j}$ that count maps by genus, vertices, and faces. These formulas give by far the fastest known way of computing these numbers, or the fixed-genus generating functions, especially for large $g$. In the very particular case of one-face maps, we recover the Harer-Zagier recurrence formula. Our main formula is a consequence of the KP equation for the generating function of bipartite maps, coupled with a Tutte equation, and it was apparently unnoticed before. It is similar in look to the one discovered by Goulden and Jackson for triangulations, and indeed our method to go from the KP equation to the recurrence formula can be seen as a combinatorial simplification of Goulden and Jackson's approach (together with one additional combinatorial trick). All these formulas have a very combinatorial flavour, but finding a bijective interpretation is currently unsolved.Comment: Version 3: We changed the title once again. We also corrected some misprints, gave another equivalent formulation of the main result in terms of vertices and faces (Thm. 5), and added complements on bivariate generating functions. Version 2: We extended the main result to include the ability to track the number of faces. The title of the paper has been changed accordingl

Asymptotic enumeration of non-crossing partitions on surfaces

We generalize the notion of non-crossing partition on a disk to general surfaces with boundary. For this, we consider a surface S and introduce the number CS(n) of noncrossing partitions of a set of n points laying on the boundary of SPostprint (author's final draft

Asymptotic enumeration and limit laws for graphs of fixed genus

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