Problems in Enumerative Combinatorics and Applications

枚举组合学及其应用中的问题

• 批准号: 1161414
• 负责人: Huafei Yan
• 金额: $ 15万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1161414&HistoricalAwards=false
• 关键词: Problems; Enumerative; Combinatorics; Applications

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.

该研究项目是关于枚举和代数组合学的，这是当代数学中快速发展的领域。 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，设定分区，整数序列，一般图形以及Ferrers板和其他多支着的填充物。在整个数学中，研究某些感兴趣的对象的一种常见和成功的方法是研究这些对象的功能。在组合学中，这种功能通常称为组合统计。该研究项目强调了组合统计的代数特性及其与其他数学分支的联系。 主要主题包括（1）为等分分配的组合统计之间的插值家族开发一个代数理论，（2）研究下降集和各种组合结构的下降多项式及其与一个依赖性确定点过程的新兴领域的关系，（3）表征跨层构造中的构造和嵌套构造中的构造构造构成构造的构图，（3）生物学。拟议的研究位于组合学的中心，这是关于离散结构的存在，枚举，分析和优化的数学分支。研究者使用组合的代数和组合方法来研究许多离散结构的特性和统计数据，并将连贯性和统一带入组合术的学科。拟议的研究的结果与代数，群体代表理论，数字理论和概率理论等数学领域具有连接和应用，以及更多应用的主题，尤其是计算机科学，图形编码和绘图，VLSI设计和数学生物学。