Several Problems On Real-Rooted Polynomials In Combinatorics

To study combinatorial properties of discrete sequences by means of zeros of generating functions is an important topic in combinatorics.This thesis is devoted to the study of several problems on real-rooted polynomials in combinatorics.Contents of the thesis are organized as follows:The first chapter is an introduction to the present research of real-rooted polynomials in combinatorics,and it also includes the relative notations,basic terminology and the main methods used later.In the second chapter,we characterize the relations between locations and multiplicities of zeros of a sequence of real-rooted polynomials defined by a three-term recurrence relation.As applications,we can improve the results of B(?)na and Wilf about alternating runs,and several well-known facts are also followed.In the third chapter,we derive some basic properties of the generating function of the joint distribution of excedances,fixed points and cycles over the symmetric group.We first give the reality of zeros,summation formula and explicit formula of q-Eulerian polynomials,and then give a combinatorial interpretation of the explicit formula.Finally,we investigate the joint distribution of excedances and cycles on derangements.Our main results improve and generalize some recent results of Brenti, Br(a|¨)nd(?)n,Mantaci,Rakotondrajao,Zhang and so on.In the fourth chapter,based on the method of zeros interlacing,we show that the generating functions of {1,m}-compositions form a Sturm sequence.Thus we resolve a conjecture of B(?)na.By the same technique,we can show that the generating functions of the sequence of the binomial coefficients related to generalized Lucas numbers form a Sturm sequence as well as the generating functions of the sequence of the associated r-Stirling numbers.In the fifth chapter,applying the technique of compatible polynomials,we inves- tigate a conjecture of Haglund,Ono and Wagner on monotone column permanent. We also give a unified approach to the fact that rook polynomials and hit polynomials of any Ferrers boards are real-rooted.