Some Properties of a Class of Refined Eulerian Polynomials  

作　　者：Yidong SUN Liting ZHAI 

机构地区：[1]School of Science, Dalian Maritime University

出　　处：《Journal of Mathematical Research with Applications》2019年第6期593-602,共10页数学研究及应用（英文版）

基　　金：Supported by “Liaoning Bai Qian Wan Talents Program” and by the Fundamental Research Funds for the Central Universities(Grant No.3132019323)

摘　　要：Recently, Sun defined a new kind of refined Eulerian polynomials, namely,An(p,q)=∑π∈■np^odes(π)q^edes(π) for n ≥ 1, where Sn is the set of all permutations on {1, 2,..., n}, odes(π) and edes(π) enumerate the number of descents of permutation π in odd and even positions, respectively. In this paper,we obtain an exponential generating function for An(p, q) and give an explicit formula for An(p, q)in terms of Eulerian polynomials An(q) and C(q), the generating function for Catalan numbers.In certain cases, we establish a connection between An(p, q) and An(p, 0) or An(0, q), and express the coefficients of An(0, q) by Eulerian numbers An,k. Consequently, this connection discovers a new relation between Euler numbers En and Eulerian numbers An,k.Recently, Sun defined a new kind of refined Eulerian polynomials, namely,■for n ≥ 1, where Sn is the set of all permutations on {1, 2,..., n}, odes（π） and edes（π） enumerate the number of descents of permutation π in odd and even positions, respectively. In this paper,we obtain an exponential generating function for An（p, q） and give an explicit formula for An（p, q）in terms of Eulerian polynomials An（q） and C（q）, the generating function for Catalan numbers.In certain cases, we establish a connection between An（p, q） and An（p, 0） or An（0, q）, and express the coefficients of An（0, q） by Eulerian numbers An,k. Consequently, this connection discovers a new relation between Euler numbers En and Eulerian numbers An,k.

关 键 词：EULERIAN polynomial EULERIAN NUMBER Euler NUMBER DESCENT alternating PERMUTATION CATALAN NUMBER 

分 类 号：O17[理学—数学]