Papers and Publications
Repetitions of Pak-Stanley Labels in G-Shi Arrangements (2022)
Given a simple graph G, one can define a hyperplane arrangement called the G-Shi arrangement. The Pak-Stanley algorithm labels the regions of this arrangement with G-parking functions. When G is a complete graph, the Pak-Stanley labels give a bijection with ordinary parking functions. However, for proper subgraphs G ⊂ Kn, while the Pak-Stanley labels still include every G-parking function, they may repeat a parking function multiple times. These repetitions are a topic of interest in the study of G-Shi arrangements and G-parking functions. Furthermore, G-parking functions are connected to many other combinatorial objects (for example, superstable configurations in chip-firing). The key insight of our work is the introduction of a combinatorial model called the "Three Rows Game". Analyzing the histories of this game and the ways in which they can induce the same outcomes allows us to characterize the repetitions in the Pak-Stanley labels.
Paper to be submitted. ArXiv preprint available here.
Around Tokuyama's Formula (2021)
Tokuyama’s formula offers a link between combinatorics and representation theory. Namely, it interprets an expression involving the characters of general linear groups as being a sum over one of three combinatorial objects: Gelfand-Tsetlin patterns, shifted tableaux, or gamma ice models. We first review existing literature concerning Tokuyama’s formula and then present two novel proofs which avoid complicated machinery required by previous proofs. We have describe progress in extending our results toward an analogous combinatorial identity for the characters of symplectic groups.
Paper unsubmitted, but available here.
Petal Projections, Knot Colorings, and Determinants (2020)
An übercrossing diagram is a knot diagram with only one crossing that may involve more than two strands of the knot. Such a diagram without any nested loops is called a petal projection. Every knot has a petal projection from which the knot can be recovered using a permutation that represents strand heights. Using this permutation, we give an algorithm that determines the p-colorability and the determinants of knots from their petal projections. In particular, we compute the determinants of all prime knots with crossing number less than 10 from their petal permutations.
An übercrossing diagram is a knot diagram with only one crossing that may involve more than two strands of the knot. Such a diagram without any nested loops is called a petal projection. Every knot has a petal projection from which the knot can be recovered using a permutation that represents strand heights. Using this permutation, we give an algorithm that determines the p-colorability and the determinants of knots from their petal projections. In particular, we compute the determinants of all prime knots with crossing number less than 10 from their petal permutations.