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

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 $\mathfrak{S}(n)$ in terms of free cumulants $R_2, R_3, \dots$ of the corresponding Young diagram. Our interpretation is based on counting certain factorizations of a given permutation.