Some Properties of a Class of Refined Eulerian Polynomials

Abstract:

      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 $A_{n,k}$.