Unimodality Problems In Combinatorics

Unimodality problem is one of the primary topics in combinatorics,including the study of unimodality,log-concavity,log-convexity and P(?)lya frequency property. Since P(?)lya frequency sequences have much better behavior than unimodal and logconcave sequences,it may often be more convenient to prove that a sequence is P(?)lya frequency.On the other hand,many unimodal and log-concave sequences arising in combinatorics are actually P(?)lya frequency sequences.So it is natural to pay more attention to P(?)lya frequency sequences,as well as polynomials having only real zeros. This thesis is devoted to the reality of zeros of polynomial sequences and the logconvexity of combinatorial sequences.The main frame is as follows.The first part of the thesis studies polynomial sequences to have only real zeros. There have been a lot of classical results about polynomials with only real zeros in the literature,but we will focus on the reality of zeros of polynomial sequences satisfying certain recurrence relations in a unified manner.Based on the method of interlacing zeros,we develop techniques for dealing with a sequence of polynomials to have only real zeros.As applications,several well-known facts are followed,including the reality of zeros of orthogonal polynomials,matching polynomials,Narayana polynomials,Bell polynomials and Eulerian polynomials.We also give the sufficient conditions for real polynomial matrices preserving the interlacing.With these results,we can settle certain conjectures of Stahl on genus polynomials and solve his open problems.The second part investigates the log-convexity of combinatorial sequences.Log-convexity is formally equivalent to the log-concavity of the reciprocal sequence,but it is true that they are fundamentally different.First,we show various operators on sequences that preserve log-convexity,such as,componentwise sum,binomial convolution, and the linear transformations given by the matrices of binomial coefficients and Stirling numbers of two kinds.Second,we discuss the log-convexity of sequences satisfying a three-term recurrence.As consequences,some famous combinatorial sequences, including the Catalan numbers,the Motzkin numbers,the Fine numbers,the central Delannoy numbers and the Schr(o|¨)der numbers,are shown to be log-convex.Finally we introduce the concept of q-log-convexity and give the sufficient conditions for the q-log-convexity of polynomial sequences.We also establish the connection between q-log-convexity and linear transformations preserving the log-convexity.As applications, the q-log-convexity of Bell polynomials,Eulerian polynomials,q-Schr(o|¨)der numbers and q-central Delannoy numbers is presented.