组合序列单峰型问题研究中的概率方法

Unimodality problems are one of the most fundamental problems in many branches such as combinatorics, algebra, geometry and probability theory. They have wide applications in many fields such as computer science and economics. Probabilistic methods are now hot in mathematics, which are also a powerful tool in combinatorics. However, there is not enough study on the applications of probabilistic methods to unimodality problems. The project focuses on the unimodality problems of combinatorial sequences to investigate the applications of probabilistic methods in combinatorics. The project includes the following issues: (i) the unimodality problems of Fuss-Narayana numbers and the Stirling numbers of the second kinds based on their relations with probability distributions; (ii) the unimodality problems of posets by means of Fortuin-Kasteleyn-Ginibre inequality; (iii) the unimodality problems of the difference of two random variables. The project not only finds a new approach to unimodality problems, but also provides a new junction between combinatorics and probability theory.

单峰型问题是组合、代数、几何、概率等多个数学分支中最基本的问题之一，并在计算机科学和经济学等领域有广泛应用。概率方法是当今数学中的研究热点，也是组合数学研究中的强有力工具。而概率方法对单峰型问题研究的应用尚处于起始阶段。本项目以组合序列的单峰型问题为切入点，研究概率方法在组合数学中的应用。本项目的研究内容主要包括: 将概率分布与组合序列相结合，研究Fuss-Narayana数和第二类Stirling数的单峰型问题；应用Fortuin-Kasteleyn-Ginibre不等式研究偏序集中的单峰型问题；系统研究两个随机变量之差的单峰型问题。本项目不仅能够为单峰型问题的研究找到新的研究思路，也将为组合数学与概率论之间提供新的交叉点。

本项目主要研究与组合序列的单峰型性质相关的几个问题。包括以下几点：(1) 证明广义升阶乘函数所构成的变换能够保持组合序列的PF性质，并指出广义降阶乘函数不能构成此类变换。同时得到一大类新的与两类Stirling数相关的变换矩阵；(2)证明两个独立的具有相同单峰分布的离散型随机变量之差具有对称单峰分布，从而说明离散型和连续型随机变量在该结论上的一致性；(3) 建立投票路和代数理论中与有根树相关的统计量之间的对应关系，从而肯定了F. Chapoton提出的一个猜想。本项目所取得的研究结果为单峰型问题的研究增添了新内容。