Another proof of Harer-Zagier formula

TL;DR

This paper presents a new, elementary combinatorial proof of the Harer-Zagier formula, which counts gluings of a regular 2n-gon by genus, using Fourier transforms and symmetric group characters.

Contribution

It provides an alternative, characters-based proof of the Harer-Zagier formula, avoiding Gaussian integrals and complex character calculations.

Findings

New proof using Fourier transform of probability measures on symmetric groups

Elementary combinatorial techniques complement character-based methods

Reinforces the validity of the Harer-Zagier formula through an accessible approach

Abstract

For a regular $2n$-gon there are $(2n-1)!!$ ways to match and glue the $2n$ sides. The Harer-Zagier bivariate generating function enumerates the gluings by $n$ and the genus $g$ of the attendant surface and leads to a recurrence equation for the counts of gluings with parameters $n$ and $g$. This formula was originally obtained by using the multidimensional Gaussian integrals. Soon after Jackson and later Zagier found alternative proofs that used the symmetric group characters. In this note we give a different, characters-based, proof. Its core is computing and marginally inverting Fourier transform of the underlying probability measure on $S_{2n}$. Aside from Murnaghan-Nakayama rule for one-hook diagrams, the counting techniques we use are of elementary, combinatorial nature.