The $q$-analogues of $\gamma$-positivity of Eulerian polynomials via group actions

Abstract

Han, Jouhet and Zeng established $q$-analogues of the $\gamma$-expansion formulas for Eulerian polynomials of types $A$ and $B$. Combinatorial interpretations of the corresponding coefficients$a_{n,k}(q)$ and $b_{n,k}(q)$, however, remained open. In this paper, we provide combinatorial interpretations for thesecoefficients by using the model of increasing binary trees, thereby resolving a problem posed by Han, Jouhet and Zeng. Our combinatorial approach consists of three main steps: 1. construct a Carlitz-type insertion bijection for increasing binary trees and derive a new combinatorial interpretation ofCarlitz's $q$-Eulerian polynomials of type $A$ in terms of such trees; 2. introduce a generalized Foata--Strehl action on increasing binary trees to interpret the coefficients $a_{n,k}(q)$;3. derive a new combinatorial interpretation for the $q$-Eulerianpolynomials of type $B$ introduced by Chow and Gessel in terms of increasing binary trees of type $B$, and develop a generalizedFoata--Strehl action on these trees to interpret the coefficients $b_{n,k}(q)$. We further give combinatorial interpretations for the quotients ${a_{n,k}(q)/(-q;q)_{k-1}}$ and ${b_{n,k}(q)/(1+q)^k(-q;q^2)_k}$ in terms of Andr\'e trees and a certain class of increasing binary trees of type $B$, respectively. As an application of the latter interpretation, we obtain a combinatorial interpretation for a $q$-analogue of the secant number and prove the positivity conjecture of Han, Jouhet and Zeng.