LEAPS-MPS: Algebraic and Combinatorial Methods in Permutation Enumeration

排列计数是计数和代数组合学的一个分支，涉及排列的计数：不同物体的线性排列。排列枚举中的问题通常是由其他数学分支--如代数、概率论和几何--驱动的，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。计数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示；反过来，可以利用这种代数结构来推导新的计数结果。组合数学和代数之间的这种相互作用是这个项目的第一个目标的中心，该目标是促进Hopf-代数方法在置换计数中的发展和应用。这个项目的第二个目标是为戴维森学院的学生建立一个集数学研究、专业发展和教育外展为一体的暑期体验DREAM(发现研究和扩大数学的途径)。这个项目建立在以前在排列计数、对称函数理论和组合Hopf代数的交叉点上的工作基础上。这一领域的经典结果是Gessel的游程定理，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计数具有给定游程长度的排列。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设置上，这将导致一种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交替排列和逆交替排列上的逆统计量(如逆下降数和逆峰值数)的分布。梦想计划的研究部分集中在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角色和数学社区面临的DEJ问题进行阅读。梦想参与者还将与PI和威廉·A·霍夫高中的合作者合作，为霍夫的EOS计划组织一次外展活动，该计划服务于有色人种学生和低收入学生。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.