Combinatorial Properties Of Two Families Of Generalized Eulerian Polynomials

In combinatorics,Eulerian polynomial is an extremely essential polynomial and Eulerian number is a sequence which has been studied extensively.They are of very important significance in the study and application of combinatorics.Recently,the generalization of Eulerian numbers and Eulerian polynomials becomes a very important issue.Brenti and Welker gave the definition and properties of restricted Eulerian polynomials by restricting the first letter of a permutation.These polynomials are also called j-polynomials by Petersen.In addition,Nunge defined the generalized Eulerian polynomials on segrnented permutations,and he further conjectured that the coefficients of these generalized Eulerian polynomials are unimodal.In this paper,we study two families of generalized Eulerian polynomials.Firstly,wc prove Nunge's unimodal conjecture on segmented permutations by using Boreca and Branden's multi-variate stable Eulerian polynomials.In addition,we give several combinatorial interpretations of Brenti and Welker's restricted polynomials based on the known conclusions.Besides,we further defined a new family of restricted polynomials which generalize Brenti's restricted polynomials.This paper is organized as folows:In the first chapter,we give a brief introduction of this paper.We introduce the study back-ground of Eulerian polynomia.ls,generalized Eulerian polynomials,restricted Eulerian polynomials,multivariate sta.ble polynomials and Carlitz identity.In the second chapter,we study the generalized Eulerian polynomials on segmented permu-tations defined by Nunge,and we prove the unilodality conjecture proposed by Nunge in two different ways.The first method use the theory of multivariate stable polynomials and some linear operators which preserve stablity.In the second proof,we give some recurrence relations of those generalized Eulerian polynomials,and prove their real-rootedness directly.In addition,we also de-velop a general approach to obtain generalized Sturm sequences from bivariate stable polynomials.In the third chapter,we study the restricted Eulerian polynomials defined by Brenti and Welker.On the base of that,we further define the case when the first term and the last term of a permutation both restricted and gave some recurrence relations of these new generalized Eulerian polynomials.By using of that and the definition of permanent,we gave three combinatorial inter-pretations of restricted Eulerian polynomials.We also give an interpretation with respect to the restricted Eulerian polynomials which are given by restricting the last letter of a permutation.In addition,we give a generalization of the famous Carlitz identity in the end.