Br\"and\'en's $(p,q)$-Eulerian polynomials, Andr\'e permutations and continued fractions

TL;DR

This paper provides a combinatorial interpretation of certain $(p,q)$-Eulerian polynomial coefficients using André permutations, addressing an open problem and extending Br"andén's earlier work on polynomial divisibility and $ ext{γ}$-expansions.

Contribution

It offers a new combinatorial interpretation of Br"andén's $(p,q)$-Eulerian polynomial coefficients in terms of André permutations, solving a recent open problem.

Findings

Polynomial $rac{ ext{γ}_{n,k}(p,q)}{(p+q)^k}$ is interpreted combinatorially.

Provides a combinatorial proof related to André permutations.

Addresses and resolves a recent open problem in the field.

Abstract

In 2008 Br\"and\'en proved a $(p,q)$-analogue of the $\gamma$-expansion formula for Eulerian polynomials and conjectured the divisibility of the $\gamma$-coefficient $\gamma_{n,k}(p,q)$ by $(p+q)^k$. As a follow-up, in 2012 Shin and Zeng showed that the fraction $\gamma_{n,k}(p, q)/(p + q)^k$ is a polynomial in $\N[p,q]$. The aim of this paper is to give a combinatorial interpretation of the latter polynomial in terms of Andr\'e permutations, a class of objects first defined and studied by Foata, Sch\"utzenberger and Strehl in the 1970s. It turns out that our result provides an answer to a recent open problem of Han, which was the impetus of this paper.