A curious symmetric decomposition of the (des, exc)-Eulerian polynomials

TL;DR

This paper introduces a novel symmetric decomposition of the joint distribution polynomial of descent and excedance statistics over the symmetric group, extending prior combinatorial results with a new $t$-symmetric perspective.

Contribution

It presents a new $t$-symmetric decomposition of the joint distribution polynomial, building upon and extending the classical equidistribution results of descent and excedance statistics.

Findings

Provides a $t$-symmetric decomposition of the generating polynomial

Extends the classical equidistribution of descent and excedance statistics

Offers new combinatorial insights into symmetric group statistics

Abstract

One of the most central result in combinatorics says that the descent statistic and the excedance statistic are equidistribued over the symmetric group. As a continuation of the work of Shareshian-Wachs (Adv. Math., 225(6) (2010), 2921--2966), we provide a curious $t$-symmetric decomposition for the generating polynomial of the joint distribution of the descent and excedance statistics over the symmetric group.