Barycentric subdivisions and derangement polynomials for the even-signed permutation groups

TL;DR

This paper extends the concept of derangement polynomials to even-signed permutation groups, linking combinatorial enumeration with geometric interpretations via barycentric subdivisions.

Contribution

It introduces a new derangement polynomial for even-signed permutations, demonstrating its nonnegativity, symmetry, unimodality, and providing an explicit exponential generating function.

Findings

Coefficients are nonnegative, symmetric, and unimodal.

The polynomial counts derangements by a new excedance statistic.

An explicit exponential generating function is derived.

Abstract

The derangement polynomial for the symmetric group enumerates derangements by the number of excedances. It can be interpreted as the local $h$-polynomial, in the sense of Stanley, of the barycentric subdivision of the simplex. Motivated by this interpretation, we define a derangement polynomial for the even-signed permutation group. The coefficients of this polynomial are nonnegative, symmetric and unimodal. We show that they enumerate derangements in the even-signed permutation group according to a notion of excedance, which is analogous to the one introduced by Brenti for signed permutations. We also give an explicit formula for the corresponding exponential generating function.