LEAPS-MPS: Algebraic and Combinatorial Methods in Permutation Enumeration

排列枚举是枚举学和代数组合学的一个分支，涉及排列计数：不同对象的线性排列。排列枚举中的问题通常是由数学的其他分支（如代数、概率论和几何）引起的，并应用于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。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霍普夫代数的代数对象来表示；反过来，这种代数结构可以用来推导新的枚举结果。组合学和代数之间的相互作用是本项目的第一个目标的核心，该目标是促进hopf代数方法在置换枚举中的发展和应用。该项目的第二个目标是建立DREAM（发现研究和扩展数学的途径），为戴维森学院的学生提供一个集数学研究、专业发展和教育推广为一体的暑期体验。这个项目建立在以前的工作在置换枚举，对称函数理论，和组合霍普夫代数的交集。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，它给出了一种系统的方法来枚举具有规定运行长度的排列。一个研究目标是将运行定理提升到非交换有色对称函数的集合，从而得到一种具有有色运行长度限制的有色排列计数的通用方法。另一个研究目标是研究交替置换和逆交替置换上逆统计量（如逆下降数和逆峰数）的分布。DREAM项目的研究部分侧重于排列枚举中的组合证明，学生参与者将参与围绕社区在数学中的作用和数学社区面临的DEIJ问题的阅读。DREAM的参与者还将与PI和William A. Hough高中的合作者合作，为Hough的EOS项目组织一次外展活动，该项目为有色人种和低收入学生提供服务。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.