Brändén's $(p,q)$-Eulerian polynomials, André permutations and continued fractions

Abstract

In 2008 Brändén 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 $\mathbb{N}[p,q]$. The aim of this paper is to give a combinatorial interpretation of the latter polynomial in terms of André permutations, a class of objects first defined and studied by Foata, Schützenberger 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.