Séminaire Lotharingien de Combinatoire, 84B.36 (2020), 12 pp.

Qiong Qiong Pan and Jiang Zeng

The γ-Coefficients of Brändén's (p,q)-Eulerian Polynomials and André Permutations

Abstract. In 2008 Brändén proved a (p,q)-analogue of the γ-expansion formula for Eulerian polynomials and conjectured the divisibility of the γ-coefficient γn,k(p,q) by (p+q)k. As a follow-up, in 2012 Shin and Zeng showed that the fraction γn,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 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.


Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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