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 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.
Highlights
Introduction and main resultsConsider a regular, oriented, 2n-gon
Gamburd used a Fourier transform-based bound for the total variation distance between two probability measures on a finite group, due to Diaconis and Shahshahani [4], [5], to prove that, when 2 lcm{2, k} | kn, γ is asymptotically uniform on the alternating subgroup Akn
For an even N = 2n, consider a directed polygon with N sides labeled by elements of [N ], and let α be a fixed-point free involution of [N ], i.e. a permutation of N ] with all cycles the electronic journal of combinatorics 23(1) (2016), #P1.21 of length 2
Summary
Introduction and main resultsConsider a regular, oriented, 2n-gon. There are (2n−1)!! ways to match and glue 2n-sides observing head-to-tail constraint in each glued pair. Let εg(n) denote the total number of gluings resulting in a surface of genus g.
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