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.

置换枚举是与计数排列有关的枚举和代数组合的分支：不同对象的线性排列。置换枚举中的问题通常是由数学的其他分支（例如代数，概率理论和几何学）引起的，并且在包括理论计算机科学，基因组学和统计力学等科学领域都有应用。枚举结果可以是更深的数学结构的迹象，有时可以通过称为组合Hopf代数的代数对象表示。反过来，可以探索这种代数结构以得出新的枚举结果。组合学和代数之间的这种相互作用是该项目的第一个目标的核心，即推进置换列出中Hopf-Elgebraic方法的开发和应用。该项目的第二个目标是建立梦想（发现研究并扩大了对数学的访问），戴维森学院学生的夏季经验，整合了数学研究，专业发展和教育范围。该项目基于固定枚举，对称功能理论和联合Hopf Elgebras的先前工作。该领域的一个经典结果是Gessel的Run定理，这是一种互惠公式，涉及非共同对称函数，该函数为枚举具有规定的运行长度的排列提供了系统方法。一个研究的目标是将运行定理提升为非交换性彩色对称函数的设置，这将导致一种对有色运行长度限制的有色排列的一般方法。另一个研究目的是研究替代排列和反向置换的反向统计数据（例如反下降数和反峰数）的分布。 Dream计划的研究组成部分着重于列出列举中的组合证明，学生参与者将参与围绕社区在数学和数学社区面临的DEIJ问题中的作用的读物。理想的参与者还将与William A. Hough高中的PI和合作者合作，为Hough组织EOS计划的外展活动，该活动为有色和低收入学生的学生提供服务。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子和更广泛的影响审查Criteria的评估来通过评估来诚实的支持。