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Planar maps and continued fractions
 Comm. Math. Phys
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UNIFIED BIJECTIONS FOR MAPS WITH PRESCRIBED DEGREES AND GIRTH
"... 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 in ..."
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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 rootface) of girth d having a rootface of degree d. We then obtain more general bijective constructions for annular maps (maps with two distinguished rootfaces) of girth at least d. Our bijections associate to each map a decorated plane tree, and nonroot 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 Fd(xd,xd+1,xd+2,...) of plane maps of girth d with rootface of degree d, where the variable xk counts the nonroot faces of degree k. The expression for F1 was already obtained bijectively by Bouttier, Di Francesco and Guitter, but for d ≥ 2 the expression of Fd is new. We also obtain an expression for the generating function G (d,e) p,q (xd,xd+1,...) of annular maps with rootfaces of degrees p and q, such that cycles separating the two rootfaces 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.
A bijection for triangulations, for quadrangulations, for pentagulations, etc
"... A dangulation is a planar map with faces of degree d. We present for each integer d ≥ 3 a bijection between the class of dangulations of girth 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 t ..."
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A dangulation is a planar map with faces of degree d. We present for each integer d ≥ 3 a bijection between the class of dangulations of girth 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. Bijections already existed for triangulations (d = 3) and for quadrangulations (d = 4). As a matter of fact, our construction unifies a bijection by Fusy, Poulalhon and Schaeffer for triangulations and a bijection by Schaeffer for quadrangulations. For d ≥ 5, both the bijections and the enumerative results are new. We also extend our bijections so as to enumerate pgonal dangulations, that is, dangulations with a simple boundary of length p. We thereby recover bijectively the results of Brown for pgonal triangulations and quadrangulations and establish new results for d ≥ 5. A key ingredient in our proofs is a class of orientations characterizing dangulations of girth d. Earlier results by Schnyder and by De Fraisseyx and Ossona de Mendez showed that triangulations of girth 3 and quadrangulations of girth 4 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 dangulation 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.
Scaling Limits of Random Trees and Planar Maps
 CLAY MATHEMATICS PROCEEDINGS
, 2012
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AN ANALOGUE OF THE HARERZAGIER FORMULA FOR UNICELLULAR MAPS ON GENERAL SURFACES
"... Abstract. A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is simply connected. In a famous article, Harer and Zagier established a formula for the generating function of unicellular maps counted according to the number of vertices a ..."
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Abstract. A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is simply connected. In a famous article, Harer and Zagier established a formula for the generating function of unicellular maps counted according to the number of vertices and edges. The keystone of their approach is a counting formula for unicellular maps on orientable surfaces with n edges, and with vertices colored using every color in [q] (adjacent vertices are authorized to have the same color). We give an analogue of this formula for general (locally orientable) surfaces. Our approach is bijective and is inspired by Lass’s proof of the HarerZagier formula. We first revisit Lass’s proof and twist it into a bijection between unicellular maps on orientable surfaces with vertices colored using every color in [q], and maps with vertex set [q] on orientable surfaces with a marked spanning tree. The bijection immediately implies HarerZagier’s formula and a formula by Jackson concerning bipartite unicellular maps. It also shed a new light on constructions by Goulden and Nica, Schaeffer and Vassilieva, and Morales and Vassilieva. We then extend the bijection to general surfaces and obtain a correspondence between unicellular maps on general surfaces with vertices colored using every color in [q], and maps on orientable surfaces with vertex set [q] with a marked planar submap. This correspondence gives an analogue of the HarerZagier formula for general surfaces. We also show that this formula implies a recursion formula due to Ledoux for the numbers of unicellular maps with given numbers of vertices and edges. 1.
A simple formula for the series of bipartite and quasibipartite maps with boundaries
, 2012
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Exact solution of two classes of prudent polygons
, 2009
"... Prudent walks are selfavoiding walks on a lattice which never step into the direction of an already occupied vertex. We study the closed version of these walks, called prudent polygons, where the last vertex of the walk is adjacent to its first one. More precisely, we give the halfperimeter genera ..."
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Prudent walks are selfavoiding walks on a lattice which never step into the direction of an already occupied vertex. We study the closed version of these walks, called prudent polygons, where the last vertex of the walk is adjacent to its first one. More precisely, we give the halfperimeter generating functions of two subclasses of prudent polygons on the square lattice, which turn out to be algebraic and nonDfinite, respectively.
Constellations and multicontinued fractions: application to Eulerian triangulations
"... Abstract. We consider the problem of enumerating planar constellations with two points at a prescribed distance. Our approach relies on a combinatorial correspondence between this family of constellations and the simpler family of rooted constellations, which we may formulate algebraically in terms ..."
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Abstract. We consider the problem of enumerating planar constellations with two points at a prescribed distance. Our approach relies on a combinatorial correspondence between this family of constellations and the simpler family of rooted constellations, which we may formulate algebraically in terms of multicontinued fractions and generalized Hankel determinants. As an application, we provide a combinatorial derivation of the generating function of Eulerian triangulations with two points at a prescribed distance. Résumé. Nous considérons le problème du comptage des constellations planaires à deux points marqués à distance donnée. Notre approche repose sur une correspondance combinatoire entre cette famille de constellations et celle, plus simple, des constellations enracinées. La correspondance peut être reformulée algébriquement en termes de fractions multicontinues et de déterminants de Hankel généralisés. Comme application, nous obtenons par une preuve combinatoire la série génératrice des triangulations eulériennes à deux points marqués à distance donnée.
ON TRIANGULATIONS WITH HIGH VERTEX DEGREE
, 2006
"... Abstract. We solve three enumerative problems concerning families of planar maps. More precisely, we establish algebraic equations for the generating function of nonseparable triangulations in which all vertices have degree at least d, for a certain value d chosen in {3, 4, 5}. The originality of t ..."
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Abstract. We solve three enumerative problems concerning families of planar maps. More precisely, we establish algebraic equations for the generating function of nonseparable triangulations in which all vertices have degree at least d, for a certain value d chosen in {3, 4, 5}. The originality of the problem lies in the fact that degree restrictions are placed both on vertices and faces. Our proofs first follow Tutte’s classical approach: we decompose maps by deleting the root and translate the decomposition into an equation satisfied by the generating function of the maps under consideration. Then we proceed to solve the equation obtained using a recent technique that extends the socalled quadratic method. 1.