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arXiv · 0908.1284

A new explicit formula for Kerov polynomials

P. Petrullo·D. Senato
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Abstract

We prove a formula expressing the Kerov polynomial $Σ_k$ 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 $Σ_k$ 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 $Σ_k$.

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P. Petrullo, D. Senato. 2009-08-10. A new explicit formula for Kerov polynomials. https://arxiv.org/abs/0908.1284

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