A simple formula for bipartite and quasi-bipartite maps with boundaries

Abstract

International audience We obtain a very simple formula for the generating function of bipartite (resp. quasi-bipartite) planar maps with boundaries (holes) of prescribed lengths, which generalizes certain expressions obtained by Eynard in a book to appear. The formula is derived from a bijection due to Bouttier, Di Francesco and Guitter combined with a process (reminiscent of a construction of Pitman) of aggregating connected components of a forest into a single tree. Nous obtenons une formule très simple pour la série génératrice des cartes biparties ayant des bords (trous) de tailles fixées, généralisant certaines expressions obtenues par Eynard dans un livre à paraître. Nous obtenons la formule à partir d'une bijection due à Bouttier, Di Francesco et Guitter, combinée avec un processus (dans l'esprit d'une construction due à Pitman) pour agréger les composantes connexes d'une forêt en un unique arbre.

Vertex Degrees in Planar Maps

“…Instead of investigating planar maps themselves, we will follow the principle presented in [5], whereby pointed planar maps are bijectively related to a certain class of trees called mobiles. (Their version of mobiles differ from the definition originally given in [3]; the equivalence of the two definitions is not shown explicitly in [5], but [7] gives a straightforward proof.) Definition 1.…”

“…Proof. For the proof of the bijection between mobiles and pointed maps we refer to [7], where the bipartite case is also discussed. It just remains to note that the induced bijection on the edges can be directly used to transfer the root edge together with its direction.…”

“…We start with bipartite mobiles since they are more easy to count, in particular if we consider rooted bipartite mobiles, see [7].…”

Limit laws of planar maps with prescribed vertex degrees

“…Instead of investigating planar maps themselves, we will follow the principle presented by Chapuy, Fusy, Kang, and Shoilekova in [8], whereby pointed planar maps are bijectively related to a certain class of trees called mobiles. (Their version of mobiles differs from the definition originally given in [5]; the equivalence of the two definitions is not shown explicitly in [8], but [10] gives a straightforward proof.) Definition.…”

“…Proof. For the proof of the bijection between mobiles and pointed maps we refer to [10], where the bipartite case is also discussed. It just remains to note that the induced bijection on the edges can be directly used to transfer the root edge together with its direction.…”