Bijective Counting of Maps by Girth and Degrees I: Restricted Boundary Conditions
نویسندگان
چکیده
For each positive integer d, we present a bijection between the set of planar maps of girth d inside a d-gon and a set of decorated plane trees. The bijection has the property that each face of degree k in the map corresponds to a vertex of degree k in the tree, so that maps of girth d can be counted according to the degree distribution of their faces. More precisely, we obtain for each integer d an explicit expression for the multivariate series Fd(xd, xd+1, xd+2, . . .) counting rooted maps of girth d inside a d-gon, where each variable xk marks the number of inner faces of degree k. The series F1 (corresponding to maps inside a loop) was already computed bijectively by Bouttier, Di Francesco and Guitter, but for d ≥ 2 the expression of Fd is new. As special cases, we recover several known bijections (bipartite maps, loopless triangulations, simple triangulations, simple quadrangulations, etc.). Our strategy is based on the use of a “master bijection”, introduced by the authors in a previous paper, between a class of oriented planar maps and a class of decorated trees. We obtain our bijections for maps of girth d by specializing the master bijection. Indeed, by defining some “canonical orientations” for maps of girth d, it is possible to identify the class of maps of girth d inside a d-gon with a class of oriented maps on which the master bijection specializes nicely. The same strategy was already used in a previous article in order to count d-angulations of girth d, and what we present here is a very significant extension of those results.
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