A new bijection relating $q$-Eulerian polynomials

TL;DR

This paper introduces a novel bijection on permutations that connects classical permutation statistics with those related to $q$-Eulerian polynomials, providing new insights into their combinatorial structure.

Contribution

It constructs a new bijection linking permutation statistics $(maj-exc,des,exc)$ to $(maj_2, ilde{des_2},inv_2)$, enriching the combinatorial understanding of $q$-Eulerian polynomials.

Findings

Establishes a bijection between two statistic vectors on permutations.

Provides a combinatorial interpretation of $q$-Eulerian polynomials.

Enhances understanding of permutation statistics related to chromatic quasisymmetric functions.

Abstract

On the set of permutations of a finite set, we construct a bijection which maps the 3-vector of statistics $(maj-exc,des,exc)$ to a 3-vector $(maj\_2,\widetilde{des\_2},inv\_2)$ associated with the $q$-Eulerian polynomials introduced by Shareshian and Wachs in \textit{Chromatic quasisymmetric functions, arXiv:1405.4269(2014).}