A new explicit formula for Kerov polynomials

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 Bóna and Simion. We also obtain symmetric functions specializing in the coefficients of \(\Sigma_k\).