Edgewise Subdivisions, Local $h$-Polynomials, and Excedances in the Wreath Product $\mathbb{Z}_r \wr \mathfrak{S}_n$

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 S n by the number of excedances. A generalization of this interpretation is given for the local h-polynomial of the rth edgewise subdivision of the barycentric subdivision of the simplex. This polynomial is shown to be γ-nonnegative and a combinatorial interpretation to the corresponding γ-coefficients is provided. The new combinatorial interpretations involve the notions of flag excedance and descent in the wreath product Z r ≀ S n . A related result on the derangement polynomial for Z r ≀ 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.

Symmetric Decompositions and Real-Rootedness

“…If the coefficients of p satisfy 0 ≤ p 0 ≤ p d ≤ p 1 ≤ p d−1 ≤ · · · ≤ p ⌊ d+1 2 ⌋ , we say that p is alternatingly increasing. This property, which is stronger than unimodality, has recently been the focus of a variety of conjectures in combinatorics as well as a means by which to prove unimodality [1,3,4,6,30,31]. The alternatingly increasing property of p is inherently tied to a unique symmetric decomposition of p: Every polynomial p of degree d can be uniquely decomposed as p = a + xb where a and b are symmetric with respect to d and d − 1, respectively.…”

Binomial Eulerian polynomials for colored permutations

“…Section 2 includes preliminaries on colored permutations, symmetric functions and γ-positivity and fixes notation. The first part of Theorem 1.2 is derived in Section 3 from the γ-positivity of the derangement polynomials for r-colored permutations, established in [2]. The proof yields a combinatorial interpretation for the corresponding γ-coefficients (Corollary 3.3) which generalizes the one provided by Theorem 1.1 in the case r = 1.…”

“…The flag excedance number is defined as fexc(w) = r · exc A (w) + ε 1 + ε 2 + · · · + ε n , where exc A (w) is the number of indices k ∈ [n] such that σ(k) > k and ε k = 0. More information, for instance about the generating functions of these statistics over Z ≀ S n , and references can be found in [2,Section 2]. Symmetric functions.…”

Real-rootedness of variations of Eulerian polynomials

“…The real-rootedness of derangement polynomials of type B d n,2 (z) was proved by Chen, Tang, and Zhao [12], and by Chow [13], independently. Athanasiadis [2] showed that d n,r (z) can be expressed as…”

“…One of its properties is that its faces F are divided into r dim(F ) faces of the same dimension. Athanasiadis [2,3] showed that ℓ V (2 V ) r , x = E r (x + x 2 + · · · + x r−1 ) n ,…”