LEAPS-MPS: Algebraic and Combinatorial Methods in Permutation Enumeration

LEAPS-MPS：排列枚举中的代数和组合方法

• 批准号: 2316181
• 负责人: Yan Zhuang
• 金额: $ 24.67万
• 依托单位: Davidson College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2316181&HistoricalAwards=false
• 关键词: LEAPS; MPS; Algebraic; Combinatorial; Methods

Permutation enumeration is the branch of enumerative and algebraic combinatorics concerned with counting permutations: linear arrangements of distinct objects. Questions in permutation enumeration are often motivated by other branches of mathematics—such as algebra, probability theory, and geometry—and have applications to scientific domains including theoretical computer science, genomics, and statistical mechanics. Enumerative results can be a sign of deeper mathematical structure, which sometimes can be expressed via algebraic objects called combinatorial Hopf algebras; in turn, this algebraic structure can be exploited to derive new enumerative results. This interplay between combinatorics and algebra is central to the first goal of this project, which is to advance the development and application of Hopf-algebraic methods in permutation enumeration. The second goal of this project is to establish DREAM (Discovering Research and Expanding Access to Mathematics), a summer experience for Davidson College students integrating mathematical research, professional development, and educational outreach.This project builds on previous work at the intersection of permutation enumeration, symmetric function theory, and combinatorial Hopf algebras. A classical result in this domain is Gessel’s run theorem, a reciprocity formula involving noncommutative symmetric functions which gives a systematic method for the enumeration of permutations with prescribed run lengths. One research objective is to lift the run theorem to the setting of noncommutative colored symmetric functions, which would lead to a general method for counting colored permutations with restrictions on colored run lengths. Another research objective is to study the distributions of inverse statistics (such as the inverse descent number and the inverse peak number) over alternating permutations and reverse-alternating permutations. The research component of the DREAM program focuses on combinatorial proofs in permutation enumeration, and student participants will engage in readings centered around the role of community in mathematics and DEIJ issues facing the mathematical community. DREAM participants will also work with the PI and collaborators from William A. Hough High School to organize an outreach event for the EOS program at Hough, which serves students of color and low-income students.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

排列计数是计数组合学和代数组合学的分支，涉及计数排列：不同对象的线性排列。排列枚举中的问题通常是由代数学的其他分支（如代数、概率论和几何）所激发的，并在包括理论计算机科学、基因组学和统计力学在内的科学领域中有应用。枚举结果可以是更深层次的数学结构的标志，有时可以通过称为组合Hopf代数的代数对象来表达;反过来，这种代数结构可以用来导出新的枚举结果。组合学和代数之间的这种相互作用是本项目的第一个目标的核心，即推进置换枚举中的Hopf代数方法的发展和应用。该项目的第二个目标是建立梦想（发现研究和扩大数学的访问），戴维森学院的学生整合数学研究，专业发展和教育推广的夏季经验。该项目建立在以前的工作在排列枚举，对称函数理论和组合Hopf代数的交叉。一个经典的结果在这一领域是Gessel的运行定理，互惠公式涉及非交换对称函数，它给出了一个系统的方法枚举排列规定的运行长度。一个研究目标是将游程定理提升到非交换有色对称函数的情形，这将导致一个计算有色排列的一般方法，并限制有色游程长度。另一个研究目标是研究逆统计量（如逆下降数和逆峰数）在交替排列和反向交替排列上的分布。梦想计划的研究部分侧重于排列枚举中的组合证明，学生参与者将围绕社区在数学中的作用和数学社区面临的DEIJ问题进行阅读。梦想的参与者也将与PI和合作者从威廉A。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。