LEAPS-MPS: Combinatorics from an Algebraic and Geometric Lens

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).The focus of this project is threefold. First, to study permutations from a combinatorial and geometric point of view. Permutations are ways to describe symmetries and arrangements of objects. They have applications in many fields such as computer science (e.g., sorting algorithms), physics (e.g., describing states of quantum particles), and biology (e.g., describing RNA sequences). This project seeks to describe the composition of permutations that share a given collection of properties and their spatial distribution when plotting these permutations in a Cartesian space. The second focus is the study of arithmetical structures of graphs. Graphs are ways to geometrically describe a given relationships between a collection of objects. Arithmetical structures of graphs provide (among other things) a combinatorial description of the number of times certain curves intersect. The PI seeks to describe the total number of arithmetical structures in certain collection of graphs. Finally, the project will create the Villanova-Puerto Rico Research Retreat (VPR^3), a collaboration research summer program between students at Villanova University and the University of Puerto Rico. The project builds on previous work to better understand peak and descents of permutations. This project aims to describe the structure coefficients of the peak algebra and provide a combinatorial reciprocity theory for the coefficients of peak and descent polynomials. These polynomials are crucial in the enumeration of permutations with a given peak or descent set. A second goal is to describe a collection of polytopes created by permutations that share peak sets and those that share descent sets. Finally, the PI will study the collection of arithmetical structures on a family of “Y-graphs” to enumerate them and to describe the effect of a smoothing operation on these arithmetical structures.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.