A New Class of Refined Eulerian Polynomials

In this note we introduce a new class of refined Eulerian polynomials defined by

$\begin{displaymath}A_n(p,q)=\sum_{\pi\in\mathfrak{S} _{n}}p^{{{\rm odes}\,}(\pi)}q^{{{\rm edes}\,}(\pi)},\end{displaymath}$

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.

Received January 31 2018; revised versions received May 13 2018; May 17 2018. Published in Journal of Integer Sequences, May 26 2018.

A New Class of Refined Eulerian Polynomials

Hua Sun College of Sciences Dalian Ocean University Dalian 116023 PR China

Hua Sun
College of Sciences
Dalian Ocean University
Dalian 116023
PR China