Simple recurrence formulas to count maps on orientable surfaces

TL;DR

This paper introduces simple recurrence formulas for counting rooted orientable maps by edges, genus, vertices, and faces, significantly improving computational efficiency especially for large genus, and connects these formulas to the KP equation and Tutte equations.

Contribution

It presents the first simple recurrence formulas for counting maps on orientable surfaces, derived from the KP and Tutte equations, with a combinatorial approach similar to Goulden and Jackson's.

Findings

Fastest known computation method for map counts

Recovers Harer-Zagier recurrence for one-face maps

Links recurrence formulas to KP and Tutte equations

Abstract

We establish a simple recurrence formula for the number $Q_g^n$ of rooted orientable maps counted by edges and genus. We also give a weighted variant for the generating polynomial $Q_g^n(x)$ where $x$ is a parameter taking the number of faces of the map into account, or equivalently a simple recurrence formula for the refined numbers $M_g^{i,j}$ that count maps by genus, vertices, and faces. These formulas give by far the fastest known way of computing these numbers, or the fixed-genus generating functions, especially for large $g$. In the very particular case of one-face maps, we recover the Harer-Zagier recurrence formula. Our main formula is a consequence of the KP equation for the generating function of bipartite maps, coupled with a Tutte equation, and it was apparently unnoticed before. It is similar in look to the one discovered by Goulden and Jackson for triangulations, and indeed our method to go from the KP equation to the recurrence formula can be seen as a combinatorial simplification of Goulden and Jackson's approach (together with one additional combinatorial trick). All these formulas have a very combinatorial flavour, but finding a bijective interpretation is currently unsolved.