代数组合学中的若干问题

发布时间：2018-05-08 14:28

本文选题：Sperner定理 + 交偏序集　；参考：《大连理工大学》2016年博士论文

【摘要】：代数组合学是组合数学中一个重要的研究内容,它主要是运用代数学中的方法或结论来研究组合数学中的问题.本文分别从凸集的Sperner性质、超平面配置的超级可解性以及Riordan矩阵行多项式矩阵的组合性质等方面着手研究了代数组合学中的若干问题.具体内容如下：第一部分研究了Sperner定理在凸集上的推广.Sperner定理主要研究子集格上的Sperner性质,是偏序集上的经典结论之一.Sperner性质在其他偏序集上的推广研究是组合数学中一个极为活跃的课题.Akiyama和Frankl猜想一般凸集上也有Sperner性质.本部分证明Akiyama-Frankl猜想在一些经典凸集上成立,例如降簇、Lih簇以及压缩理想.第二部分考虑超平面配置的超级可解性.如果超平面配置对应的交偏序集存在极大的模链则该超平面配置称为超级可解的.Stanley证明了图配置超级可解的充分必要条件是该图是一个弦图.本文受图配置超级可解充分必要条件的启发,研究了推广的图配置即ψ图配置超级可解的等价条件.超平面配置的自由性可以推出超级可解性,而对于图配置而言两者是等价的,即图配置的超级可解性也可推出自由性.本部分考虑了ψ图的自由性,并给出ψ图配置自由性的必要条件.第三部分研究了Riordan矩阵行多项式矩阵的组合性质.首先给出了Riordan矩阵行多项式矩阵的两种等价刻画,随后研究了其各种组合性质,包括行多项式矩阵的q-全正性、第0列的q-对数凸性以及每行的q-对数凹性.本文着重研究了Bell型Riordan矩阵和Aigner-Riordan矩阵的行多项式矩阵,并给出判断行多项式矩阵全正性的条件.作为应用,从行多项式矩阵的角度将一些已知的零散的结果以统一的形式给出.Barry提出了三个关于幂级数系数的Hankel行列式的猜想,通过计算移位Aigner-Riordan数的Hankel行列式统一地证明了Barry提出的三个猜想.
[Abstract]:Algebraic combinatorics is an important research content in combinatorial mathematics. It mainly uses methods or conclusions in algebra to study problems in combinatorial mathematics. In this paper, some problems in algebraic combinatorics are studied in terms of the Sperner properties of convex sets, the super solvability of hyperplane configurations and the combinatorial properties of Riordan matrix row polynomial matrices. The main contents are as follows: in the first part, the generalization of Sperner theorem on convex sets is studied. The Sperner theorem mainly studies the Sperner properties on subset lattices. The generalization of the Sperner property on other partial ordered sets is a very active subject in combinatorial mathematics. Akiyama and Frankl conjecture that the general convex sets also have Sperner properties. In this part, we prove that Akiyama-Frankl 's conjecture holds on some classical convex sets, such as reduced cluster Lih clusters and contractive ideals. In the second part, we consider the super solvability of hyperplane configuration. If there is a maximal module chain in the intersection partial order set corresponding to the hyperplane configuration, then the hyperplane configuration is called super solvable. Stanley proves the necessary and sufficient condition for a graph to be super solvable if and only if the graph is a chord graph. In this paper, inspired by the necessary and sufficient conditions for the super solvability of graph configuration, we study the equivalent conditions of generalized graph configuration, that is, 蠄 graph configuration super solvability. The supersolvability of hyperplane configuration can be deduced, but the two are equivalent to graph configuration, that is, the super solvability of graph configuration can also deduce freedomness. In this part, we consider the freeness of 蠄 graphs and give the necessary conditions for the configuration freeness of 蠄 graphs. In the third part, the combinatorial properties of row polynomial matrices of Riordan matrices are studied. In this paper, two equivalent characterizations of row polynomial matrices of Riordan matrices are given. Then, we study the combination properties of row polynomial matrices, including the q-total positivity of row polynomial matrices, the q-logarithmic convexity of column 0 and the q-logarithmic concave of each row. In this paper, the row polynomial matrices of Bell type Riordan matrices and Aigner-Riordan matrices are studied, and the conditions for judging the full positivity of row polynomial matrices are given. As an application, from the point of view of row polynomial matrix, some known scattered results are given in a unified form. Barry puts forward three conjectures about the Hankel determinant of power series coefficients. Three conjectures proposed by Barry are proved uniformly by calculating the Hankel determinant of shift Aigner-Riordan number.
【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O157

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