Generic method for bijections between blossoming trees and planar maps

This article presents a unified bijective scheme between planar maps and blossoming trees, where a blossoming tree is defined as a spanning tree of the map decorated with some dangling half-edges that enable to reconstruct its faces. Our method generalizes a previous construction of Bernardi by loosening its conditions of applications so as to include annular maps, that is maps embedded in the plane with a root face different from the outer face. The bijective construction presented here relies deeply on the theory of \alpha-orientations introduced by Felsner, and in particular on the existence of minimal and accessible orientations. Since most of the families of maps can be characterized by such orientations, our generic bijective method is proved to capture as special cases all previously known bijections involving blossoming trees: for example Eulerian maps, m-Eulerian maps, non separable maps and simple triangulations and quadrangulations of a k-gon. Moreover, it also permits to obtain new bijective constructions for bipolar orientations and d-angulations of girth d of a k-gon. As for applications, each specialization of the construction translates into enumerative by-products, either via a closed formula or via a recursive computational scheme. Besides, for every family of maps described in the paper, the construction can be implemented in linear time. It yields thus an effective way to encode and generate planar maps. In a recent work, Bernardi and Fusy introduced another unified bijective scheme, we adopt here a different strategy which allows us to capture different bijections. These two approaches should be seen as two complementary ways of unifying bijections between planar maps and decorated trees.Comment: 45 pages, comments welcom

Dissections, orientations, and trees, with applications to optimal mesh encoding and to random sampling

We present a bijection between some quadrangular dissections of an hexagon and unrooted binary trees, with interesting consequences for enumeration, mesh compression and graph sampling. Our bijection yields an efficient uniform random sampler for 3-connected planar graphs, which turns out to be determinant for the quadratic complexity of the current best known uniform random sampler for labelled planar graphs [{\bf Fusy, Analysis of Algorithms 2005}]. It also provides an encoding for the set $\mathcal{P}(n)$ of $n$-edge 3-connected planar graphs that matches the entropy bound $\frac1n\log_2|\mathcal{P}(n)|=2+o(1)$ bits per edge (bpe). This solves a theoretical problem recently raised in mesh compression, as these graphs abstract the combinatorial part of meshes with spherical topology. We also achieve the {optimal parametric rate} $\frac1n\log_2|\mathcal{P}(n,i,j)|$ bpe for graphs of $\mathcal{P}(n)$ with $i$ vertices and $j$ faces, matching in particular the optimal rate for triangulations. Our encoding relies on a linear time algorithm to compute an orientation associated to the minimal Schnyder wood of a 3-connected planar map. This algorithm is of independent interest, and it is for instance a key ingredient in a recent straight line drawing algorithm for 3-connected planar graphs [\bf Bonichon et al., Graph Drawing 2005]

Characters and conjugacy classes of the symmetric group

Rapport interne.This article addresses several conjectures due to Jacob Katriel concerning conjugacy classes of $\S{n}$ viewed as operators acting by multiplication. The first conjecture expresses, for a fixed partition \r of the form $r1^{n-r}$, the eigenvalues (or central characters) \eo\r\l in terms of contents of \l. While Katriel conjectured a generic form and an algorithm to compute missing coefficients, we provide an explicit expression. The second conjecture (presented at FPSAC'98 in Toronto) gives a general form for the expression of a conjugacy class in terms of \emph{elementary} operators. We prove it using a convenient description by differential operators acting on symmetric polynomials. To conclude, we partially extend our results on \eo\r\l to arbitrary partitions \r. || Nous démontrons plusieurs conjectures dues à Jacob Katriel qui portent sur les classes de conjugaisons de $\S{n}$ vues comme opérateurs agissant par multiplication. La première conjecture exprime, pour une partition fixée $\rho$ de la forme $r1^{n-r}$, l

Enfants conçus avec une aide médicale dans la cohorte Elfe

Objectif : étudier les conditions entourant la naissance des enfants conçus avec une aide médicale, et explorer le rôle des facteurs sociaux et médicaux dans la prématurité et l’hypotrophie.Méthodes : l'analyse porte sur 9495 enfants de la cohorte Elfe, de terme supérieur à 33 semaines d'aménorrhée, singletons ou jumeaux, répartis en trois groupes selon leur contexte de conception, "traitement", "infertiles non traités" et "fertiles" (790, 1044 et 7661 enfants). Cinq champs ont été décrits (caractéristiques sociales, antécédents maternels, grossesse, naissance, santé à deux mois). Une analyse univariée et multivariée des facteurs susceptibles d'être associés à la prématurité ou à l’hypotrophie a été réalisée. Les sous-groupes "fécondation in vitro" et "stimulation ovarienne" ont été comparés.Résultats : le groupe "traitement" présente une situation sociale très favorable, des grossesses plus compliquées et une gémellité, une prématurité et une hypotrophie significativement plus fréquentes. La gémellité et la prématurité sont significativement plus fréquentes en FIV qu'en stimulation. Après ajustement, il existe une association significative entre contexte de conception et prématurité, sans différence entre groupes traitement et infertile ni entre groupes FIV et stimulation. Aucune association n’est retrouvée avec l’hypotrophie.Conclusion : ces résultats suggèrent un sur-risque de complications néonatales chez les enfants de parents infertiles, traités ou non. Des particularités du groupe "infertile" empêchent cependant d'exclure un effet des techniques médicales. Le projet se poursuivra avec l'analyse du développement psychomoteur et pondéral entre la naissance et 3 ans

Succinct representation of triangulations with a boundary

We consider the problem of designing succinct geometric data structures while maintaining efficient navigation operations. A data structure is said succinct if the asymptotic amount of space it uses matches the entropy of the class of structures represented. For the case of planar triangulations with a boundary we propose a succinct representation of the combinatorial information that improves to 2.175 bits per triangle the asymptotic amount of space required and that supports the navigation between adjacent triangles in constant time (as well as other standard operations). For triangulations with $m$ faces of a surface with genus g, our representation requires asymptotically an extra amount of 36(g-1)lg m bits (which is negligible as long as g << m/lg m)

A random tunnel number one 3-manifold does not fiber over the circle

We address the question: how common is it for a 3-manifold to fiber over the circle? One motivation for considering this is to give insight into the fairly inscrutable Virtual Fibration Conjecture. For the special class of 3-manifolds with tunnel number one, we provide compelling theoretical and experimental evidence that fibering is a very rare property. Indeed, in various precise senses it happens with probability 0. Our main theorem is that this is true for a measured lamination model of random tunnel number one 3-manifolds. The first ingredient is an algorithm of K Brown which can decide if a given tunnel number one 3-manifold fibers over the circle. Following the lead of Agol, Hass and W Thurston, we implement Brown's algorithm very efficiently by working in the context of train tracks/interval exchanges. To analyze the resulting algorithm, we generalize work of Kerckhoff to understand the dynamics of splitting sequences of complete genus 2 interval exchanges. Combining all of this with a "magic splitting sequence" and work of Mirzakhani proves the main theorem. The 3-manifold situation contrasts markedly with random 2-generator 1-relator groups; in particular, we show that such groups "fiber" with probability strictly between 0 and 1.Comment: This is the version published by Geometry & Topology on 15 December 200

On square permutations

Severini and Mansour introduced in [4]square polygons, as graphical representations of square permutations, that is, permutations such that all entries are records (left or right, minimum or maximum), and they obtained a nice formula for their number. In this paper we give a recursive construction for this class of permutations, that allows to simplify the derivation of their formula and to enumerate the subclass of square permutations with a simple record polygon. We also show that the generating function of these permutations with respect to the number of records of each type is algebraic, answering a question of Wilf in a particular case

Random trees between two walls: Exact partition function

We derive the exact partition function for a discrete model of random trees embedded in a one-dimensional space. These trees have vertices labeled by integers representing their position in the target space, with the SOS constraint that adjacent vertices have labels differing by +1 or -1. A non-trivial partition function is obtained whenever the target space is bounded by walls. We concentrate on the two cases where the target space is (i) the half-line bounded by a wall at the origin or (ii) a segment bounded by two walls at a finite distance. The general solution has a soliton-like structure involving elliptic functions. We derive the corresponding continuum scaling limit which takes the remarkable form of the Weierstrass p-function with constrained periods. These results are used to analyze the probability for an evolving population spreading in one dimension to attain the boundary of a given domain with the geometry of the target (i) or (ii). They also translate, via suitable bijections, into generating functions for bounded planar graphs.Comment: 25 pages, 7 figures, tex, harvmac, epsf; accepted version; main modifications in Sect. 5-6 and conclusio