Research

arXiv:2609.00202 

A quiver representation consists of a collection of vector spaces along with a set of arrows, which are linear maps between these spaces. In this work, we study quiver representations in equioriented type A which are also chain complexes; that is, in which consecutive arrows compose to zero. We show that orbits of these representations under a change of basis action are in bijection with mixed dimer covers of a 2×n grid graph. The latter object can be endowed with a partial order which is a distributive lattice, and we show that the degeneration order on the orbits of chain complexes is a coarsening of this partial order. In addition, we use recent matrix formulae of Claussen and Ovenhouse to enumerate these orbits. This also computes the Kostant partition function applied to height-restricted, type A roots. When the dimension vector is uniform, we discuss a correspondence with paths of a beam of light bouncing between glass plates and give an explicit generating function.

arXiv:2608.06338 

The symmetric group S_n is generated by transpositions, and problems of sorting permutations using transpositions are well studied. In recent work, Adin, Alon, and Roichman studied the related problem of sorting n points on a circle, and gave a formula for the maximum number of adjacent swaps required. This is equivalent to the number of adjacent transpositions required to transform any permutation into a power of the cyclic permutation (1, 2, …, n).
The focus of this work is an analogous question in the alternating group An, which is generated by 3-cycles. That is, using 3-cycles instead of transpositions, what is the maximum number of steps required to transform an even permutation into a power of (1, 2, …, n) in the alternating group? We determine this number exactly for even n and n ≡ 1 (mod 4). For n ≡ 3 (mod 4), we show that the sorting number can take one of two possible values and give explicit constructions demonstrating that the larger value occurs infinitely often.

arXiv:2412.16820

For integral weights λ and μ of a classical simple Lie algebra 𝔤, Kostant's weight multiplicity formula gives the multiplicity of the weight μ in the irreducible representation with highest weight λ, which we denote by m(λ,μ). Kostant's weight multiplicity formula is an alternating sum over the Weyl group of the Lie algebra whose terms are determined via a vector partition function. The Weyl alternation set (λ,μ) is the set of elements of the Weyl group that contribute nontrivially to the multiplicity m(λ,μ). In this article, we prove that Weyl alternation sets are order ideals in the weak Bruhat order of the corresponding Weyl group. Specializing to the Lie algebra 𝔰𝔩_{r+1}(ℂ), we give a complete characterization of the Weyl alternation sets (α̃ ,μ), where α̃  is the highest root and μ is a negative root, answering a question of Harry posed in 2024. We also provide some enumerative results that pave the way for our future work, where we aim to prove Harry's conjecture that the q-analog of Kostant's weight multiplicity formula is mq(α̃ ,μ)=qr+j−i+1+qr+j−i−qj−i+1 when μ=−(α_i+α_{i+1}+⋯+α_j) is a negative root of 𝔰𝔩_{r+1}(ℂ).

arXiv:2508.13917

In a parking function, a car is considered lucky if it is able to park in its preferred spot. Extending work of Harris and Martinez, we enumerate outcomes of parking functions with a fixed set of lucky cars. We then consider a generalization of parking functions known as vector parking functions or u-parking functions, in which a nonnegative integer capacity is given to each parking spot in the street. With certain restrictions on u, we enumerate outcomes of u-parking functions with a fixed set of lucky cars or with a fixed number of lucky cars. We also count outcomes according to which spots contain lucky cars, and give formulas for enumerating u-parking functions themselves according to their set of lucky cars.

arXiv:2410.22432 

Starting with an inclusion-exclusion proof of a combinatorial identity, a direct bijection can be produced using recursive subtraction (sometimes with a direct combinatorial description). We apply this method to identities for generalized Wilf equivalences among consecutive patterns in inversion sequences, giving direct bijective proofs of some generalized Wilf equivalences shown by Auli and Elizalde. We also give new bijective proofs of a stronger relation among some consecutive patterns.

arXiv:2301.13718

Poster from Permutation Patterns 2022 "Recursive maps for derangements and nonderangements" can be viewed here.

A derangement is a permutation with no fixed point, and a nonderangement is a permutation with at least one fixed point. There is a one-term recurrence for the number of derangements of n elements, and we describe a bijective proof of this recurrence which can be found using a recursive map. We then show the combinatorial interpretation of this bijection and how it compares with other known bijections, and show how this gives an involution on 𝔖_n. Nonderangements satisfy a similar recurrence. We convert the bijective proof of the one-term identity for derangements into a bijective proof of the one-term identity for nonderangements.