Some Properties of a Class of Refined Eulerian Polynomials
Received:March 30, 2019  Revised:September 04, 2019
Key Word: Eulerian polynomial   Eulerian number   Euler number   descent   alternating permutation   Catalan number
Fund ProjectL:Supported by Liaoning BaiQianWan Talents Program" and by the Fundamental Research Funds for the Central Universities (Grant No.3132019323).
 Author Name Affiliation Yidong SUN School of Science, Dalian Maritime University, Liaoning 116026, P. R. China Liting ZHAI School of Science, Dalian Maritime University, Liaoning 116026, P. R. China
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Recently, Sun defined a new kind of refined Eulerian polynomials, namely, $$A_n(p,q)=\sum_{\pi\in \mathfrak{S}_n}p^{{\rm odes}(\pi)}q^{{\rm edes}(\pi)}$$ for $n\geq 1$, where $\mathfrak{S}_n$ is the set of all permutations on $\{1, 2, \dots, n\}$, $\mathrm{odes}(\pi)$ 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 $A_{n,k}$. Consequently, this connection discovers a new relation between Euler numbers $E_n$ and Eulerian numbers $A_{n,k}$.