A new class of refined Eulerian polynomials

Abstract

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

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