  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.
