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A New Class of Refined Eulerian Polynomials Hua Sun¹
College of Sciences
Dalian Ocean University
Dalian 116023
P. R. China
mailto:sunhua@dlou.edu.cnsunhua@dlou.edu.cn

in

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.