A new explicit formula for Kerov polynomials

We prove a formula expressing the Kerov polynomial ${\mathrm{\Sigma }}_k$ as a weighted sum over the lattice of noncrossing partitions of the set $\{1,\mathrm{.}\mathrm{.}\mathrm{.},k+1\}$. In particular, such a formula is related to a partial order $\text{mirr}$ 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 ${\mathfrak{S}}_k$ and allows us to recover the coefficients of ${\mathrm{\Sigma }}_k$ 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 ${\mathrm{\Sigma }}_k$.