Some properties of a class of refined Eulerian polynomials
Received:March 30, 2019  Revised:June 28, 2019
Key Word: Eulerian polynomial   Eulerian number   Euler number   Descent   Alternating permutation   Catalan number.  
Fund ProjectL:Liaoning BaiQianWan Talents Program
Author NameAffiliationE-mail
Yidong SUN School of Science 
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      Recently, H. Sun defined a new kind of refined Eulerian polynomials, namely, \begin{eqnarray*} A_n(p,q)=\sum_{\pi\in \mathfrak{S}_n}p^{{\rm odes}(\pi)}q^{{\rm edes}(\pi)} \end{eqnarray*} for $n\geq 1$, where $\mathrm{odes}(\pi)$ and $\mathrm{edes}(\pi)$ enumerate the number of descents of permutation $\pi$ in odd and even positions, respectively. In this paper, we obtain an exponential generating function for $A_{n}(p,q)$ and give an explicit formula for $A_{n}(p,q)$ in terms of Eulerian polynomials $A_{n}(q)$ and $C(q)$, the generating function for Catalan numbers. In certain cases, we establish a connection between $A_{n}(p,q)$ and $A_{n}(p,0)$ or $A_{n}(0,q)$, and express the coefficients of $A_{n}(0,q)$ by Eulerian numbers. Consequently, this connection discovers a new relation between Euler numbers and Eulerian numbers.
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