【作者】 张华军；

【导师】 王军；

【作者基本信息】 大连理工大学 ， 计算数学， 2006， 博士

【摘要】 Sperner理论是组合数学的一个分支，其研究对象是偏序集，主要考虑偏序集上满足某些条件的极值问题，它的起源可以追朔到1928年Sperner的一个定理：在子集格中，最大秩集构成基数最大的反链．经过近一个世纪的发展，Sperner定理已经发展成为一门系统的理论。 本文第一章是Sperner理论的一个简单的综述，包括相关的记号、术语以及在后面需要用到的主要方法． 第二章给出q-阶对数凹性的定义，并把一系列保持对数凹性的线性变换推广到q-阶对数凹性。 第三章给出加权偏序集q-直积的定义，并证明关于偏序集的正规匹配性的q直积定理，即：若两个偏序集都是q-阶对数凹的并且具有正规匹配性，那么它们的每一个q-直积都是都是q-阶对数凹的并且具有正规匹配性。利用这个定理得到了子空间格的几个子偏序集是q-阶对数凹的并且具有正规匹配性。 第四章讨论子集格的四个与Lih猜想相关的子偏序集，然后通过子集格的对称链分解导出它们的套链分解。 第五章考虑置换偏序集B(n，n)的LYM性质和局部EKR性质。更多还原

【Abstract】 Sperner Theory is one of research branches in combinatorics, whose reserch object is posets, whose main content is to investgate the extremal problems on posets. Its source came from a theorem of Sperner in 1928: in a subset lattice, a size maximal rank set forms a size maximal antichain. After about one century’s development Sperner’s theorem has become into a systematic theory.The first chapter is a simple survey on this theory, including the relative notations, basic terminology and the main methods used later.In the second chapter, we introduce the definition of the q-degree log-concavity of a sequence, and give a series of linear transformations which preserve the q-degree log-concavity of a sequence.In the third chapter, we first give the definition of q-direct product, and then deduce the q-direct product theorem from product theorem: if (Q, v) and (P, w) are both q-degree log-concave and have the normalized matching property, then each q-direct product of them is q-degree log-concave and has the normalized matching property. By this theorem we prove that some subposets of L(V) are q-degree log-concave and have the normalized matching property.In the fourth chapter, we construct the nested chain decompositions for four subposets of the Boolean lattice.In the last chapter, we consider the LYM property and the local EKR property of the partial permutation poset B(n,n).更多还原