We compute, for each genus $g$ ≥ 0, the generating function $L$<sub>$g$</sub> ≡ $L$<sub>$g$</sub>($t$;$p$<sub>1</sub>,$p$<sub>2</sub>,...) 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 $g$, $L$<sub>$g$</sub> is a rational function in the new variables, computable by an explicit recursion on the genus. The same holds for the generating function $L$<sub>$g$</sub> of <i>rooted</i> 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 strengthens recent results of Kazarian and Zograf, who studied the case where the number of faces is bounded, in the equivalent formalism of <i>dessins d’enfants</i>. 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.

Source : oai:HAL:hal-01337789v1

Volume: DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)

Section: Proceedings

Published on: January 1, 2015

Submitted on: November 21, 2016

Keywords: Topological recursion,Enumeration,Maps on surfaces,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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