A new class of refined Eulerian polynomials

Abstract

In this note we introduce a new class of refined Eulerian polynomials defined by $$A_n(p,q)=\sum_{\pi\in\mathfrak{S}_n}p^{{\rm odes}(\pi)}q^{{\rm edes}(\pi)},$$ where ${\rm odes}(\pi)$ and ${\rm edes}(\pi)$ enumerate the number of descents of permutation $\pi$ in odd and even positions, respectively. We show that the refined Eulerian polynomials $A_{2k+1}(p,q),k=0,1,2,\ldots,$ and $(1+q)A_{2k}(p,q),k=1,2,\ldots,$ have a nice symmetry property.