Some Properties of a Class of Refined Eulerian Polynomials

DOI：10.3770/j.issn:2095-2651.2019.06.006

 作者 单位 孙怡东 大连海事大学理学院, 辽宁 大连 116026 翟丽婷 大连海事大学理学院, 辽宁 大连 116026

最近,孙华定义了一类新的精细化Eulerian多项式,即$$A_n(p,q)=\sum_{\pi\in \mathfrak{S}_n}p^{{\rm odes}(\pi)}q^{{\rm edes}(\pi)},\ \ n\ge 1,$$ 其中$S_n$表示$\{1,2,\ldots,n\}$上全体$n$阶排列的集合, odes$(\pi)$与edes$(\pi)$分别表示$S_n$中排列$\pi$的奇数位与偶数位上降位数的个数.本文利用经典的Eulerian多项式$A_n(q)$ 与Catalan 序列的生成函数$C(q)$,得到精细化Eulerian 多项式$A_n(p,q)$的指数型生成函数及$A_n(p,q)$的显示表达式.在一些特殊情形,本文建立了$A_n(p,q)$与$A_n(0,q)$或$A_n(p,0)$之间的联系,并利用Eulerian数表示多项式$A_n(0,q)$的系数.特别地,这些联系揭示了Euler数$E_n$与Eulerian数$A_{n,k}$之间的一种新的关系.

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 $A_{n,k}$.