Bijective counting of maps by girth and degrees I: restricted boundary conditions

Olivier Bernardi (1) , Eric Fusy (2)

1 MIT - Massachusetts Institute of Technology
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

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 $F_d(x_d,x_{d+1},x_{d+2},\ldots)$ counting rooted maps of girth~$d$ inside a $d$-gon, where each variable $x_k$ marks the number of inner faces of degree $k$. The series $F_1$ (corresponding to maps inside a loop) was already computed bijectively by Bouttier, Di~Francesco and Guitter, but for $d\geq 2$ the expression of $F_d$ 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.

Domaines: Combinatoire [math.CO]

Olivier Bernardi, Eric Fusy. Bijective counting of maps by girth and degrees I: restricted boundary conditions. 2011. ⟨hal-00566762v1⟩