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

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.