Edgewise subdivisions, local $h$-polynomials and excedances in the wreath product $\ZZ_r \wr \mathfrak{S}_n$

Abstract

The coefficients of the local $h$-polynomial of the barycentric subdivision of the simplex with $n$ vertices are known to count derangements in the symmetric group ${\mathfrak{S}}_n$ by the number of excedances. A generalization of this interpretation is given for the local $h$-polynomial of the $r$th edgewise subdivision of the barycentric subdivision of the simplex. This polynomial is shown to be $\gamma$-nonnegative and a combinatorial interpretation to the corresponding $\gamma$-coefficients is provided. The new combinatorial interpretations involve the notions of flag excedance and descent in the wreath product $\ZZ_r \wr \mathfrak{S}_n$. A related result on the derangement polynomial for $\ZZ_r \wr \mathfrak{S}_n$, studied by Chow and Mansour, is also derived from results of Linusson, Shareshian and Wachs on the homology of Rees products of posets.

Final version

References in corpus (2)

• The symmetric and unimodal expansion of Eulerian polynomials via continued fractions
• On the excedance number of colored permutation groups

Cited by in corpus (2)

• Symmetric unimodal expansions of excedances in colored permutations
• The local $h$-polynomial of the edgewise subdivision of the simplex