Topological expansion of the coefficients of zonal polynomials in genus one

TL;DR

This paper derives an explicit formula for the coefficients of zonal polynomials in genus one using a combinatorial interpretation involving unoriented maps.

Contribution

It introduces a novel combinatorial approach to compute genus one coefficients of zonal polynomials, expanding understanding of their structure.

Findings

Explicit formula for genus one coefficients

Combinatorial interpretation involving unoriented maps

Enhanced computational methods for zonal polynomials

Abstract

We use a combinatorial interpretation of the coefficients of zonal Kerov polynomials as a number of unoriented maps to derive an explicit formula for the coefficients in genus one.