New formulas counting one-face maps and Chapuy's recursion

TL;DR

This paper introduces a new formula for counting one-face maps of a given genus using involutions on permutations, leading to a derivation of Chapuy's recursion and a combinatorial interpretation of the Harer-Zagier recurrence.

Contribution

It presents a novel convolution formula involving Stirling numbers, connects it to Chapuy's recursion, and offers a combinatorial interpretation of the Harer-Zagier recurrence.

Findings

New convolution formula for $A(n,g)$ involving Stirling numbers

Derivation of Chapuy's recursion from the new formula

Combinatorial interpretation of the Harer-Zagier recurrence

Abstract

In this paper, we begin with the Lehman-Walsh formula counting one-face maps and construct two involutions on pairs of permutations to obtain a new formula for the number $A(n,g)$ of one-face maps of genus $g$. Our new formula is in the form of a convolution of the Stirling numbers of the first kind which immediately implies a formula for the generating function $A_n(x)=\sum_{g\geq 0}A(n,g)x^{n+1-2g}$ other than the well-known Harer-Zagier formula. By reformulating our expression for $A_n(x)$ in terms of the backward shift operator $E: f(x)\rightarrow f(x-1)$ and proving a property satisfied by polynomials of the form $p(E)f(x)$, we easily establish the recursion obtained by Chapuy for $A(n,g)$. Moreover, we give a simple combinatorial interpretation for the Harer-Zagier recurrence.