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

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. This scheme 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 the family of unicellular 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.