Abstract
We give a different presentation of a recent bijection due to Chapuy and Dołe ̨ga for nonorientable bipartite quadrangulations and we extend it to the case of nonorientable general maps. This can be seen as a Bouttier–Di Francesco–Guitter-like generalization of the Cori–Vauquelin–Schaeffer bijection in the context of general nonori- entable surfaces. In the particular case of triangulations, the encoding objects take a particularly simple form and we recover a famous asymptotic enumeration formula found by Gao.
Highlights
1.1 MotivationThe study of maps has seen tremendous developments in the past few decades
The case of a more general compact orientable surface has been studied, mostly in the context of uniform quadrangulations: partial convergence has been established in a series of papers ending with [Bet14] and a full convergence is under investigation [BM15b]
We give an alternate description of their bijection, which provides an explicit construction for pointed quadrangulations(i) and we show how to generalize it to nonorientable general maps
Summary
The study of maps has seen tremendous developments in the past few decades. One of the reasons is that they provide natural discrete versions of a given surface. Extending the Cori–Vauquelin–Schaeffer bijection, a combinatorial interpretation of this fact in the orientable case was given by Chapuy, Marcus and Schaeffer [CMS09] Their approach rely on a bijection between bipartite quadrangulations (it is a classical simple fact that bipartite quadrangulations are in bijection with general maps) and one-face maps of the same surface, whose vertices carry integer labels satisfying some local constraints. He studied the algebraicity of the generating function of rooted maps on a given surface with face degree constraints [Gao93] In their recent work [CD15], Chapuy and Dołega extended the construction of [CMS09] to bipartite nonorientable quadrangulations. In the very particular case of quadrangulations, the global constraints can be expressed as local constraints and the encoding objects (so called well-labeled unicellular maps) take a simple form and this allowed Chapuy and Dołega [CD15] to give a combinatorial interpretation of Bender and Canfield’s asymptotic formula. In the particular case of triangulations, the same miracle occurs and we are able to give a combinatorial interpretation of the results of [Gao91]
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