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.

排列枚举是涉及排列计数的枚举和代数组合学的一个分支：排列枚举中的问题通常由其他数学分支（例如代数、概率论和几何）激发，并应用于科学领域。包括理论计算机科学、基因组学和统计力学，枚举结果可以是更深层次数学结构的标志，有时可以通过称为组合 Hopf 的代数对象来表达。反过来，可以利用这种代数结构来得出新的枚举结果。组合数学和代数之间的相互作用是该项目的首要目标的核心，即推进 Hopf 代数方法在排列枚举中的开发和应用。该项目的第二个目标是建立 DREAM（发现研究和扩大数学获取途径），为戴维森学院的学生提供集数学研究、专业发展、该项目建立在排列枚举、对称函数理论和组合 Hopf 代数交叉领域的先前工作的基础上。该领域的一个经典结果是 Gessel 运行定理，这是一个涉及非交换对称函数的互易公式，它给出了一种系统方法。具有规定游程长度的排列的枚举是将游程定理提升到非交换彩色对称函数的设置，这将导致另一个研究目标是研究交替排列和反向交替排列上的逆统计量（例如逆下降数和逆峰值数）的分布。 DREAM 项目的重点是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以及数学社区面临的 DEIJ 问题进行阅读。还与 PI 和 William A. Hough 高中的合作者合作，为 Hough 的 EOS 项目组织了一次外展活动，该项目为有色人种学生和低收入学生提供服务。该奖项反映了 NSF 的法定使命，并被认为值得支持通过使用基金会的智力优点和更广泛的影响审查标准进行评估。