Problems in Enumerative Combinatorics and Applications

这个研究计画是关于计数与代数组合学，这是当代数学中发展最快的领域。PI探索了各种组合结构上的一系列相互关联的枚举问题，包括排列，匹配，集合划分，整数序列，一般图，以及费勒斯板和其他polyominoes的填充。在整个数学中，研究某些感兴趣的对象的一种常见而成功的方法是研究这些对象上的函数。在组合数学中，这样的函数通常被称为组合统计。该研究项目强调组合统计的代数性质及其与其他数学分支的联系。 主要内容包括：（1）建立等分布组合统计中插值族的代数理论，（2）研究各种组合结构的下降集和下降多项式及其与单相依行列式点过程的关系，（三）描述组合结构中的交叉和嵌套及其在图优化和计算数学中的应用生物学。所提出的研究是在组合学的中心，组合学是数学的一个分支，涉及离散结构的存在、枚举、分析和优化。研究者使用组合代数和组合方法来研究许多离散结构的属性和统计，并为组合学学科带来连贯性和统一性。建议的研究结果有联系和应用等经典领域的数学代数，群表示理论，数论和概率论，以及更多的应用学科，特别是计算机科学，图形编码和绘图，超大规模集成电路设计，数学生物学。

This research project is on enumerative and algebraic combinatorics, a fast developing area in contemporary mathematics. The PI explores a series of interrelated enumeration problems on various combinatorial structures, including permutations, matchings, set partitions, integer sequences, general graphs, and fillings of Ferrers boards and other polyominoes. Throughout mathematics a common and successful approach to studying some objects of interest is to study functions on those objects. In combinatorics such functions are often called combinatorial statistics. This research project emphasizes on the algebraic properties of combinatorial statistics and their connections to other branches of mathematics. The major topics include(1) developing an algebraic theory for the interpolating families between equidistributed combinatorial statistics,(2) studying descent sets and descent polynomials for various combinatorial structures and their relations to the emerging field of one-dependent determinantal point processes,(3) characterizing the crossing and nestings in combinatorial structures with applications in graph optimizations and computational mathematical biology.The proposed research is in the center of combinatorics, which is a branch of mathematics concerning the existence, enumeration, analysis, and optimization of discrete structures. The investigator uses a combined algebraic and combinatorial approach to investigate properties and statistics of many discrete structures, and brings coherence and unity to the discipline of combinatorics. Results of the proposed research have connections and applications to such classical areas of mathematics as algebra, group representation theory, number theory, and probability theory, as well as more applied subjects, notably computer science, graph encoding and drawing, VLSI design, and mathematical biology.