一类精细化Eulerian多项式的若干性质

最近,孙华定义了一类新的精细化Eulerian多项式,即$$A_n(p,q)=sum_{piin mathfrak{S}_n}p^{{rm odes}(pi)}q^{{rm edes}(pi)}, nge 1,$$ 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.

Some Properties of a Class of Refined Eulerian Polynomials

Recently, Sun defined a new kind of refined Eulerian polynomials, namely, $$A_n(p,q)=sum_{piin mathfrak{S}_n}p^{{rm odes}(pi)}q^{{rm edes}(pi)}$$ for $ngeq 1$, 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 $A_{n,k}$.