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

New bijective links on planar maps via orientations

This article presents new bijections on planar maps. At first a bijection is established between bipolar orientations on planar maps and specific "transversal structures" on triangulations of the 4-gon with no separating 3-cycle, which are called irreducible triangulations. This bijection specializes to a bijection between rooted non-separable maps and rooted irreducible triangulations. This yields in turn a bijection between rooted loopless maps and rooted triangulations, based on the observation that loopless maps and triangulations are decomposed in a similar way into components that are respectively non-separable maps and irreducible triangulations. This gives another bijective proof (after Wormald's construction published in 1980) of the fact that rooted loopless maps with $n$ edges are equinumerous to rooted triangulations with $n$ inner vertices.Comment: Extended and revised journal version of a conference paper with the title "New bijective links on planar maps", which appeared in the Proceedings of FPSAC'08, 23-27 June 2008, Vi\~na del Ma

Maps and trees

We present bijective proofs for the enumeration of planar maps and non-separable planar maps, and apply the same method to rederive the enumeration formula for self-dual maps

Bijections for generalized Tamari intervals via orientations

We present two bijections for generalized Tamari intervals, which were recently introduced by Pr\'eville-Ratelle and Viennot, and have been proved to be in bijection with rooted non-separable maps by Fang and Pr\'eville-Ratelle. Our first construction proceeds via separating decompositions on quadrangulations and can be seen as an extension of the Bernardi-Bonichon bijection between Tamari intervals and minimal Schnyder woods. Our second construction directly exploits the Bernardi-Bonichon bijection and the point of view of generalized Tamari intervals as a special case of classical Tamari intervals (synchronized intervals); it yields a trivariate generating function expression that interpolates between generalized Tamari intervals and classical Tamari intervals.Comment: 18 page

On Irreducible Maps and Slices

This volume honouring the Memory of Philippe FlajoletWe consider the problem of enumerating d-irreducible maps, i.e. planar maps whose all cycles have length at least d, and such that any cycle of length d is the boundary of a face of degree d. We develop two approaches in parallel: the natural approach via substitution where these maps are obtained from general maps by a replacement of all d-cycles by elementary faces, and a bijective approach via slice decomposition which consists in cutting the maps along shortest paths. Both lead to explicit expressions for the generating functions of d-irreducible maps with controlled face degrees, summarized in some elegant "pointing formula". We provide an equivalent description of d-irreducible slices in terms of so-called d-oriented trees. We finally show that irreducible maps give rise to a hierarchy of discrete integrable equations which include equations encountered previously in the context of naturally embedded trees

Limit of normalized quadrangulations: The Brownian map

Consider $q_n$ a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with $n$ faces. In this paper we show that, when $n$ goes to $+\infty$, $q_n$ suitably normalized converges weakly in a certain sense to a random limit object, which is continuous and compact, and that we name the Brownian map. The same result is shown for a model of rooted quadrangulations and for some models of rooted quadrangulations with random edge lengths. A metric space of rooted (resp. pointed) abstract maps that contains the model of discrete rooted (resp. pointed) quadrangulations and the model of the Brownian map is defined. The weak convergences hold in these metric spaces.Comment: Published at http://dx.doi.org/10.1214/009117906000000557 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Cut vertices in random planar maps

The main goal of this paper is to determine the asymptotic behavior of the number X n of cut-vertices in random planar maps with n edges. It is shown that X n / n ¿ c in probability (for some explicit c > 0 ). For so-called subcritical classes of planar maps (like outerplanar maps) we obtain a central limit theorem, too. Interestingly the combinatorics behind this seemingly simple problem is quite involved.Peer ReviewedPostprint (published version