Teaching
This summer, I am teaching Math 208, Matrix Algebra with Applications. Information about the course can be found on Canvas for registered students.
Research
My research interests are broadly in algebraic combinatorics, such as symmetric/quasisymmetric functions and permutation statistics.
My current project is about chromatic symmetric functions of trees, a follow-up of sorts to my paper "Generalized degree polynomials of trees" from last year (see below).
Publications & Preprints
- (November 2024) Generalized degree polynomials of trees (with Ricky Liu).
The generalized degree polynomial $\mathbf{G}_T(x,y,z)$ of a tree $T$ is an invariant introduced by Crew that enumerates subsets of vertices by size and number of internal and boundary edges. Aliste-Prieto et al. proved that $\mathbf{G}_T$ is determined linearly by the chromatic symmetric function $\mathbf{X}_T$, introduced by Stanley. We present several classes of information about $T$ that can be recovered from $\mathbf{G}_T$ and hence also from $\mathbf{X}_T$. Examples of such information include the double-degree sequence of $T$, which enumerates edges of $T$ by the pair of degrees of their endpoints, and the leaf adjacency sequence of $T$, which enumerates vertices of $T$ by degree and number of adjacent leaves. We also discuss a further generalization of $\mathbf{G}_T$ that enumerates tuples of vertex sets and show that this is also determined by $\mathbf{X}_T$.
- (October 2023) Shuffle bases and quasisymmetric power sums (with Ricky Liu).
We characterize isomorphisms between the algebra $\text{QSym}$ of quasisymmetric functions and the shuffle algebra $\text{Sh}$ of compositions. To do so, we establish a universal property for $\text{Sh}$ analogous to the one for $\text{QSym}$ described by Aguiar, Bergeron, and Sottile. We then use these results to derive characterizations of quasisymmetric power sums and study some particular examples from the literature.CV
TK.
- (October 2023) Shuffle bases and quasisymmetric power sums (with Ricky Liu).