A new class of refined Eulerian polynomials

Abstract

In this note we introduce a new class of refined Eulerian polynomials defined by $$A_n(p,q)=\sum_{π\in\mathfrak{S}_n}p^{{\rm odes}(π)}q^{{\rm edes}(π)},$$ where ${\rm odes}(π)$ and ${\rm edes}(π)$ enumerate the number of descents of permutation $π$ 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.

8 pages