Generating functions of bipartite maps on orientable surfaces

TL;DR

This paper derives explicit rational generating functions for bipartite maps on orientable surfaces of any genus, revealing connections to Hurwitz numbers and employing elementary proofs without topological recursion.

Contribution

It provides explicit rational formulas for generating functions of bipartite maps on surfaces of arbitrary genus, extending previous results and linking to Hurwitz number models.

Findings

Generating functions are rational in new variables for all genera.

Explicit recursion formulas are provided for computing these functions.

The results suggest deeper connections between bipartite maps and Hurwitz numbers.

Abstract

We compute, for each genus $g\geq 0$, the generating function $L_g\equiv L_g(t;p_1,p_2,\dots)$ of (labelled) bipartite maps on the orientable surface of genus $g$, with control on all face degrees. We exhibit an explicit change of variables such that for each $g$, $L_g$ is a rational function in the new variables, computable by an explicit recursion on the genus. The same holds for the generating function $F_g$ of rooted bipartite maps. The form of the result is strikingly similar to the Goulden/Jackson/Vakil and Goulden/Guay-Paquet/Novak formulas for the generating functions of classical and monotone Hurwitz numbers respectively, which suggests stronger links between these models. Our result complements recent results of Kazarian and Zograf, who studied the case where the number of faces is bounded, in the equivalent formalism of dessins d'enfants. Our proofs borrow some ideas from Eynard's "topological recursion" that he applied in particular to even-faced maps (unconventionally called "bipartite maps" in his work). However, the present paper requires no previous knowledge of this topic and comes with elementary (complex-analysis-free) proofs written in the perspective of formal power series.