A new explicit formula for Kerov polynomials

TL;DR

This paper introduces a new explicit formula for Kerov polynomials using noncrossing partitions, connecting combinatorial structures with algebraic expressions and symmetric functions.

Contribution

It provides the first explicit combinatorial formula for Kerov polynomials involving noncrossing partitions and related posets, linking algebraic and combinatorial perspectives.

Findings

Derived a weighted sum formula over noncrossing partitions for Kerov polynomials

Connected the formula to a partial order on irreducible noncrossing partitions

Reconstructed coefficients using pattern-avoiding permutation posets

Abstract

We prove a formula expressing the Kerov polynomial $\Sigma_k$ as a weighted sum over the lattice of noncrossing partitions of the set $\{1,...,k+1\}$. In particular, such a formula is related to a partial order $\mirr$ on the Lehner's irreducible noncrossing partitions which can be described in terms of left-to-right minima and maxima, descents and excedances of permutations. This provides a translation of the formula in terms of the Cayley graph of the symmetric group $\frak{S}_k$ and allows us to recover the coefficients of $\Sigma_k$ by means of the posets $P_k$ and $Q_k$ of pattern-avoiding permutations discovered by B\'ona and Simion. We also obtain symmetric functions specializing in the coefficients of $\Sigma_k$.