Some properties of a class of refined Eulerian polynomials
| Autor: | Sun, Yidong, Zhai, Liting |
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| Rok vydání: | 2018 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | In recent, 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 ${odes}(\pi)$ and ${edes}(\pi)$ enumerate the number of descents of permutation $\pi$ in odd and even positions, respectively. In this paper, we build an exponential generating function for $A_{n}(p,q)$ and establish 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 special case, we set up 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. Specially, this connection creates a new relation between Euler numbers and Eulerian numbers. Comment: 10 pages |
| Databáze: | arXiv |
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