Combinatorial proof for the rationality of the bivariate generating series of maps in positive genus

TL;DR

This paper provides the first combinatorial proof of the rationality of the generating series for maps in positive genus, using bijections and detailed analysis of decorated unicellular maps, simplifying previous computational proofs.

Contribution

It introduces a new combinatorial proof for the rationality of the generating series of maps in positive genus, based on bijections with decorated unicellular maps.

Findings

Established a combinatorial rationality scheme for maps in positive genus.

Provided a simpler combinatorial proof for the rationality of maps counted by edges.

Enhanced understanding of the structure of maps through bijections.

Abstract

In this paper, we give the first combinatorial proof of a rationality scheme for the generating series of maps in positive genus enumerated by both vertices and faces, which was first obtained by Bender, Canfield and Richmond in 1993 by purely computational techniques. To do so, we rely on a bijection obtained by the second author in a previous work between those maps and a family of decorated unicellular maps. Our main contribution consists in a fine analysis of this family of maps. As a byproduct, we also obtain a new and simpler combinatorial proof of the rationality scheme for the generating series of maps enumerated by their number of edges, originally obtained computationally by Bender and Canfield in 1991 and combinatorially by the second author in 2019.