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

TL;DR

This paper extends the combinatorial interpretation of local $h$-polynomials to edgewise subdivisions and wreath products, revealing their $ ext{γ}$-nonnegativity and involving flag excedance and descent statistics.

Contribution

It generalizes known results on local $h$-polynomials to $r$th edgewise subdivisions and wreath products, providing new combinatorial interpretations and $ ext{γ}$-nonnegativity proofs.

Findings

The local $h$-polynomial of the $r$th edgewise subdivision is $ ext{γ}$-nonnegative.

A combinatorial interpretation involving flag excedance and descent in wreath products is established.

Derived results connect derangement polynomials with homology of poset Rees products.

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.