A Class Of Refined Eulerian Polynomials And Their Properties

Permutation is a classical research object in Combinatorics,which is closely related to many important combinatorial structures,including lattice path,tree,non-crossing set partition,standard Yang table,01-matrix,etc.Since the landmark work of Macmahon P A,a famous combinatorial scientist in the early 20th century,the study of permutation statistics has become one of the core research topics in Combinatorics.The important statistics in the permutation include the major index,inversion,excedance and descent,among which descent in the permutation is counted by the Eulerian number.Eulerian number and Eulerian polynomials are important sequences in Combinatorics,Combinatorial Number Theory and Combinatorial Algebra.They are widely concerned and studied,and have different types of generalization.This paper further explores the refined Eulerian polynomials An(p,q),where Sn represents the full permutation of set {1,2,3…n},where odes(?)and edes(?)represent the descent number of permutation ? in odd and even positions,respectively.The recursive relation of the refined Eulerian polynomial An(p,q)is used to obtain the exponential generating function of An(p,q),and the classical Eulerian polynomial An(q)and the generating functions of Catalan sequences C(q)is further used to obtain the explicit formula of the refined Eulerian polynomial An(p,q).Meanwhile,the close relationship between the refined Eulerian polynomials An(p,q)and special case An(p,0)and An(0,q)is established.Based on the relations among Eulerian number An,k,Euler number En and coefficient an,k of An(0,q),the coefficient of An(0,q),some important formulas between them are obtained.The relationship between some special values of refined Eulerian polynomials An(p,q)and Euler Number En are considered.Finally,the recursive formula of an,k is studied and the combinatorial proof is also given.