LEAPS-MPS: Algebraic and Combinatorial Methods in Permutation Enumeration

Davidson College, Davidson NC

Investigators

Abstract

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.