Two positivity conjectures for Kerov polynomials

TL;DR

This paper introduces two new positivity conjectures related to Kerov polynomials, which connect symmetric group characters with free cumulants, extending previous positivity results.

Contribution

It proposes two novel positivity conjectures for Kerov polynomial coefficients, strengthening earlier conjectures and building on recent proofs.

Findings

Proposes two new positivity conjectures for Kerov polynomials

Connects Kerov polynomial coefficients with free cumulants

Builds on and extends previous positivity results

Abstract

Kerov polynomials express the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as polynomials in the free cumulants of the associated Young diagram. We present two positivity conjectures for their coefficients. The latter are stronger than the positivity conjecture of Kerov-Biane, recently proved by Feray.