排列统计及其相关计数多项式的若干问题研究

排列统计是组合计数理论中内容最丰富的课题。本项目拟借鉴Species理论和“陈氏文法”理论中相对成熟的研究方法和手段，从新的角度研究排列统计量alternating runs以及与Jacobi椭圆函数相关的排列统计量。重点刻画统计量alternating runs在D型Coxeter群、Simsun排列以及对合上的分布问题。寻找新的平台用来研究alternating runs是我们的研究目标之一。如何用与Jacobi 椭圆函数相关的微分系统刻画与三角函数相关的排列统计量是本项目重点考虑的问题。注意到Jacobi椭圆函数、正切函数、正割函数以及对应的微分多项式都与统计量峰值密切相关，挖掘这一事实隐含的数学思想也是我们的研究目标之一。研究过程中涉及到的算子既包括普通的微分算子，也包括nested导子、Jordan导子以及Schwarzian导子。最后，本项目还将研究重集圈结构的计数问题。

Permutation statistics are the richest topic in enumerative combinatorics. In this project, we shall study alternating runs of permutations and permutation statistics associated with Jacobi elliptic functions by referring to the theory of species and the grammatical method of Chen. We mainly study the distribution of alternating runs of Coxeter groups of type D, involutions and Simsun permutations. One purpose of this project is to study alternating runs via new research platforms. An important problem in this project is how to study several combinatorial structures associated with trigonometric functions by using differential systems associated with Jacobi elliptic functions. Note that Jacobi elliptic functions, tangent function, secant function and the associated derivative polynomials are all closely related to peaks of permutations. Therefore, another purpose of this project is to explore the underlying idea of this fact. In the discussion, we shall use the ordinary differential operator, nested derivative, Jordan derivative and Schwarzian derivative. Moreover, we shall also study cyclic structures of multisets.