A bijection for nonorientable general maps

TL;DR

This paper presents a new bijective approach to encode nonorientable general maps, extending existing bijections and enabling enumeration results for nonorientable surfaces, including triangulations.

Contribution

It introduces a novel bijection for nonorientable maps that generalizes previous bijections and simplifies enumeration in special cases like triangulations.

Findings

Provides a new bijection for nonorientable bipartite quadrangulations.

Extends the bijection to general nonorientable maps.

Recovers Gao's asymptotic enumeration formula for triangulations.

Abstract

We give a different presentation of a recent bijection due to Chapuy and Dol\k{e}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 nonorientable surfaces. In the particular case of triangulations, the encoding objects take a particularly simple form and this allows us to recover a famous asymptotic enumeration formula found by Gao.