组合学中单峰对称序列的研究

The unimodality of sequences is one of the primary branches of combinatorics. The unimodality of symmetric sequences is different from that of normal sequences, then a systematic study of the unimodality of symmetric sequences is expected and necessary. To posets as a platform, by the methods of linear algebra and abstract algebra respectively, the project will systematically study the unimodality of symmetric sequences and their generating functions. The main contents are as follows:.1. Polynomials of degree no more than n form a linear space. We will research the deomposition problems of symmetric polynomials, and establish the relationship between the properties of coefficients under some important bases of symmetric polynomials and unimodality or stronger properties. .2. To posets as a platform and by dint of the rich contents of the theory of posets, we systematically study the unimodality of symmetric sequences. On the one hand it is expected to constructively prove the unimodality of rank functions of some important posets. On the other hand we study the inverse problem, in order to give a constructive proof of the unimodality of a given symmetric polynomial, we need to construct a proper poset and its decomposition..3. We study the unimodality of generating functions of diverse symmetrical statistics of combinatorial objects, especially permutations and partitions, by the methods of posets and abstract algebra respectively. We will focus on the conjecture of Guo junwei and Zeng jiang that the generating functions of descent statistics on the set of involutions have non-negative r vector.

序列的单峰性问题是组合学基本研究内容之一。对称序列单峰性的研究有别于一般序列单峰性的研究，故有望且有必要进行系统的研究。本项目将分别以偏序集为平台、线性代数及抽象代数等方法系统地研究对称序列及其发生函数的单峰性质。主要内容如下：.1.次数不超过n的实系数对称多项式组成了一个线性空间。研究对称多项式的分解问题，建立对称多项式在一些重要基底表示下系数的性质与单峰性或更强性质之间的联系。.2.以偏序集为平台，借助偏序集理论丰富的内容系统地研究对称多项式的单峰性。一方面预期给出一些重要偏序集秩生成函数单峰性的构造性证明，另一方面研究其反问题，给定一个对称多项式，构造恰当的偏序集及分解进而给出对称多项式单峰性构造性的证明。.3.分别从偏序集角度及抽象代数方法研究置换及分拆等组合对象各类对称统计量生成函数的单峰性。着重研究郭军伟和曾江关于对合置换的下降统计量生成函数具有非负r向量的猜想。