【作者】 刘丽；

【导师】 王毅；

【作者基本信息】 大连理工大学 ， 基础数学， 2008， 博士

【摘要】 单峰型问题是组合学中基本的研究课题之一,其内容包括单峰性、对数凹性、对数凸性和PF性质的研究等。因PF性质蕴涵单峰性和对数凹性,且有限PF序列可由其发生函数只具有实零点来刻画,故多项式实零点性的研究是单峰型问题中的一个重要的组成部分(这项研究本身也是数学学科中的经典问题之一)。因此关于PF性质和对数凸性的研究是单峰型问题中的主要内容。本文分别统一地给出多项式序列具有实零点性以及组合序列具有对数凸性的判别方法。具体内容如下。第一部分研究多项式序列只具有实零点的问题。虽然在以往的研究过程中已经有许多关于多项式实零点性的经典结果,但是本文旨在对递归多项式的实零点性作出统一的处理。基于零点交替的方法给出递归的多项式序列只具有实零点的充分条件,根据这些条件能由已知的只具有实零点的多项式得到新的只具有实零点的多项式,所建立的方法能统一许多经典问题的证明,包括正交多项式、图的匹配多项式、Narayana多项式、Bell多项式和Eulerian多项式等的实零点性;还给出多项式矩阵保持交替性的充分条件,作为应用能证明Stahl关于某些图的亏格多项式的实零点性的猜想并能解决其提出的公开问题。第二部分对组合序列的对数凸性做出了较为系统的研究。虽然序列的对数凸性等同于其倒数序列的对数凹性,但是对数凸性与对数凹性的研究之间有着本质的区别。首先给出保持对数凸性的算子,包括对应项加和、二项式卷积以及由二项式系数、两类Stirling数决定的线性变换等;其次给出满足三项递归关系的序列具有对数凸性的充分条件,由此能够得到许多重要的组合序列的对数凸性,包括Catalan数、Motzkin数、Fine数、中心Delannoy数和两类Schr(o|¨)der数等的对数凸性;最后引入了q-对数凸性的定义,并得到多项式序列具有q-对数凸性的充分条件,还建立了q-对数凸性和保持对数凸性的线性变换之间的联系,作为应用能够得到Bell多项式、Eulerian多项式、q-Schr(o|¨)der数和q-中心Delannoy数等组合学中经典的多项式序列的q-对数凸性。更多还原

【Abstract】 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.更多还原