Research
My research interests lie in the field of Algebraic Combinatorics. Some of my favorite combinatorial objects include:
the Kronecker coefficients
vacillating tableaux
My research interests lie in the field of Algebraic Combinatorics. Some of my favorite combinatorial objects include:
the Kronecker coefficients
vacillating tableaux
Properties of the plactic monoid centralizers (with Bruce Sagan), submitted (2025)
Let $u$ be a word over the positive integers $\bbP$. Motivated by a question involving crystal graphs, Sagan and Wilson initiated the study of the centralizer of $u$ in the plactic monoid which is the set C(u) = \{w \mid \text{$uw$ is Knuth equivalent to $wu$}\}. In particular, they conjectured the following stability phenomenon: for any $u$ there is a positive integer $K$ depending only on $u$ such that $C(u^k) = C(u^K)$ for $k\ge K$. We prove that this property holds for various $u$ including words consisting of only ones and twos, as well as permutations. Sagan and Wilson also considered $c_{n,m}(u)$ which is the number of $w\in C(u)$ of length $n$ and maximum at most $m$. They showed that $c_{n,m}(1)$ is a polynomial in $m$ of degree $n-1$ and conjectured properties of the coefficients when it is expanded in a binomial coefficient basis. We prove some of these conjectures, for example, that the coefficients are always nonnegative integers.
On Partitiona associated with elementary symmetric polynomials (with Cristina Ballantine, Shaheen Nazir, Bridget Eileen Tenner and Karlee Westrem), Ramanujan J 69, 29 (2026). https://doi.org/10.1007/s11139-025-01307-z
The elementary symmetric partition function is a map on the set of integer partitions. It sends a partition to the partition whose parts are the summands in the evaluation of the elementary symmetric function on the parts of . These elementary symmetric partition functions have been studied before, and are related to plethysm. In this note, we study properties of the elementary symmetric partition functions, particularly related to injectivity and the number of parts appearing in their image partitions.
On the Correspondence Between Integer Sequences and Vacillating Tableaux (with Zhanar Berikkyzy, Pamela E. Harris, Anna Pun and Catherine Yan), Enumerative Combinatorics and Applications, 5:1 (2025), Artical #S2R7. doi: 10.54550/ECA2025V5S1R7
A fundamental identity in the representation theory of the partition algebra is $n^k = \sum_{\lambda} f^\lambda m_k^\lambda$ for $n \geq 2k$, where $\lambda$ ranges over integer partitions of $n$, $f^\lambda$ is the number of standard Young tableaux of shape $\lambda$, and $m_k^\lambda$ is the number of vacillating tableaux of shape lambda and length 2k. Using a combination of RSK insertion and jeu de taquin, Halverson and Lewandowski constructed a bijection $DI_n^k$ that maps each integer sequence in $[n]^k$ to a pair of tableaux of the same shape, where one is a standard Young tableau and the other is a vacillating tableau. In this paper, we study the fine properties of Halverson and Lewandowski's bijection and explore the correspondence between integer sequences and the vacillating tableaux via the map $DI_n^k$ for general integers $n$ and $k$. In particular, we characterize the integer sequences $\bsy{i}$ whose corresponding shape, $\lambda$, in the image $DI_n^k(i)$, satisfies $\lambda_1 = n$ or $\lambda_1 = n -k$.
The Newton polytope of the Kronecker product (with Greta Panova), Combinatorial Theory, 5(3) (2025). doi:10.5070/C65365558
We study the Kronecker product of two Schur functions $s_\lambda\ast s_\mu$, defined as the image of the characteristic map of the product of two $S_n$ irreducible characters. We prove special cases of a conjecture of Monical--Tokcan--Yong that its monomial expansion has a saturated Newton polytope. Our proofs employ the Horn inequalities for positivity of Littlewood-Richardson coefficients and imply necessary conditions for the positivity of Kronecker coefficients.
On the Kronecker product of Schur functions of square shapes, Algebraic Combinatorics, Volume 7 (2024) no. 5, pp. 1575-1600. doi: 10.5802/alco.381.
Motivated by the Saxl conjecture and the tensor square conjecture, which states that the tensor squares of certain irreducible representations of the symmetric group contain all irreducible representations, we study the tensor squares of irreducible representations associated with square Young diagrams. We give a formula for computing Kronecker coefficients, which are indexed by two square partitions and a three-row partition, specifically one with a short second row and the smallest part equal to 1. We also prove the positivity of square Kronecker coefficients for particular families of partitions, including three-row partitions and near-hooks.
Combinatorial identities for vacillating tableaux (with Zhanar Berikkyzy, Pamela E. Harris, Anna Pun and Catherine Yan), Integers, 24A (2024), 36 pages.
Vacillating tableaux are sequences of integer partitions that satisfy specific conditions. The concept of vacillating tableaux stems from the representation theory of the partition algebra and the combinatorial theory of crossings and nestings of matchings and set partitions. In this paper, we further investigate the enumeration of vacillating tableaux and derive multiple combinatorial identities and integer sequences relating to the number of vacillating tableaux, simplified vacillating tableaux, and limiting vacillating tableaux.
On the limiting vacillating tableaux for integer sequences (with Zhanar Berikkyzy, Pamela E. Harris, Anna Pun and Catherine Yan), Journal of Combinatorics Volume 15 (2024) Number 3, pp. 383-400. doi:10.4310/JOC.240907013731
A fundamental identity in the representation theory of the partition algebra is $n^k = \sum_{\lambda} f^\lambda m_k^\lambda$ for $n \geq 2k$, where $\lambda$ ranges over integer partitions of $n$, $f^\lambda$ is the number of standard Young tableaux of shape $\lambda$, and $m_k^\lambda$ is the number of vacillating tableaux of shape $\lambda$ and length $2k$.
Using a combination of RSK insertion and jeu de taquin, Halverson and Lewandowski constructed a bijection $DI_n^k$ that maps each integer sequence in $[n]^k$ to a pair consisting of a standard Young tableau and a vacillating tableau. In this paper, we show that for a given integer sequence $\bsy{i}$, when $n$ is sufficiently large, the vacillating tableaux determined by $DI_n^k(\bsy{i})$ become stable when $n \rightarrow \infty$; the limit is called the limiting vacillating tableau for $\bsy{i}$. We give a characterization of the set of limiting vacillating tableaux and present explicit formulas that enumerate those vacillating tableaux.
UC Berkeley Combinatorics Seminar, Berkeley, CA. Feb 2026.
ICERM Computation in Representation Theory workshop, Providence, RI. Nov 2025.
ICERM Machine Learning Seminar, Providence, RI. Sep 2025.
UC Davis ADM Seminar, Davis, CA. Jan 2025.
The Newton polytope of the Kronecker product, FPSAC 2024, Bochum, Germany. July 2024. (poster, extended abstract)
UCLA Student Combinatorics Seminar, Los Angeles, CA. May 2024.
USC Combinatorics Seminar, Los Angeles, CA. Feb 2024.
UC Davis ADM Seminar, Davis, CA. Jan 2024.
SoCal Discrete Math Symposium 2023, Los Angeles, CA. Nov 2023.
UCLA Combinatorics Forum, Los Angeles, CA. Oct 2023.
The Kronecker product of Schur functions, USC Combinatorics Seminar, Los Angeles, CA. Mar 2023. (video)
On the limiting vacillating tableaux for integer sequences, JMM 2023, AMS Special Session on Research Community in Algebraic Combinatorics, Boston, MA. Jan 2023.
Oregon State University Algebra and Number Theory Seminar, Corvallis, OR. Nov 2022. (video)