Flag Descents and Eulerian Polynomials for Wreath Product Quotients

TL;DR

This paper explores Eulerian polynomials and descent statistics in wreath product quotients, revealing palindromic properties and introducing a new combinatorial tool called the colored winding number.

Contribution

It introduces a new combinatorial approach to analyze flag descents and Eulerian polynomials in wreath product quotients, proving their palindromic nature.

Findings

Eulerian polynomials can be computed over wreath product quotients.

Flag descent polynomials are palindromic in these quotients.

A new combinatorial tool, the colored winding number, is introduced.

Abstract

We investigate the $\alpha$-colored Eulerian polynomials and a notion of descents introduced in a recent paper of Hedmark and show that such polynomials can be computed as a polynomial encoding descents computed over a quotient of the wreath product $\mathbb{Z}_\alpha\wr\mathfrak{S}_n$. Moreover, we consider the flag descent statistic computed over this same quotient and find that the flag Eulerian polynomial remains palindromic. We prove that the flag descent polynomial is palindromic over this same quotient by giving a combinatorial proof that the flag descent statistic is symmetrically distributed over the collection of colored permutations with fixed last color by way of a new combinatorial tool, the colored winding number of a colored permutation. We conclude with some conjectures, observations, and open questions.