  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.
