Another proof of the Harer-Zagier formula

作者：Pittel Boris^*

来源：Electronic Journal of Combinatorics, 2016, 23(1): P1.21.

DOI：10.37236/5420

• surfaces
• chord diagrams
• genus
• random permutations
• Fourier transform
• irreducible characters
• Murnaghan-Nakayama
• generating functions

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 using multidimensional Gaussian integrals. Soon after, Jackson and later Zagier found alternative proofs using symmetric group characters. In this note we give a different, characters -based, proof. Its core is computing and marginally inverting the Fourier transform of the underlying probability measure on S-2n. A key ingredient is the Murnaghan-Nakayama rule for the characters associated with one-hook Young diagrams.

• 出版日期2016-2-5