Problems in Enumerative Combinatorics and Applications

本研究项目是关于当代数学中一个发展迅速的领域--枚举和代数组合学。PI探索了各种组合结构上的一系列相互关联的计数问题，包括排列、匹配、集合划分、整数序列、一般图以及Ferrers板和其他多项式的填充。在整个数学中，研究一些感兴趣的对象的一个常见且成功的方法是研究这些对象上的函数。在组合学中，这样的函数通常被称为组合统计量。这项研究的重点是组合统计学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量之间插值族的代数理论，(2)研究各种组合结构的下降集和下降多项式，以及它们与新兴的单依赖行列式点过程领域的关系，(3)刻画组合结构中的交叉和嵌套及其在图优化和计算数学生物学中的应用。该研究处于组合学的中心，是关于离散结构的存在、计数、分析和优化的数学分支。研究人员使用代数和组合相结合的方法来研究许多离散结构的性质和统计，并使组合学的学科具有一致性和统一性。所提出的研究结果与代数、群表示论、数论和概率论等经典数学领域以及更多的应用学科，特别是计算机科学、图形编码与绘图、VLSI设计和数学生物学都有联系和应用。

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.