Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations

TL;DR

This paper provides a combinatorial interpretation of Kerov character polynomial coefficients, linking them to permutation factorizations, and thereby clarifies the structure of symmetric group characters in terms of free cumulants.

Contribution

It introduces an explicit combinatorial interpretation of Kerov character polynomial coefficients using permutation factorizations, advancing understanding of symmetric group characters.

Findings

Coefficients of Kerov polynomials correspond to counts of permutation factorizations.

The interpretation connects free cumulants to permutation structures.

Provides a new combinatorial perspective on symmetric group characters.

Abstract

We find an explicit combinatorial interpretation of the coefficients of Kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups S(n) in terms of free cumulants R_2,R_3,... of the corresponding Young diagram. Our interpretation is based on counting certain factorizations of a given permutation.