Teaching

Research

Publications & Preprints

• (May 2026) On the chromatic symmetric functions of trees (Ph.D. thesis).
  The chromatic symmetric function (CSF) of a graph, introduced by Richard Stanley in 1995, is a symmetric function generalization of the chromatic polynomial which enumerates proper colorings by the number of uses of each color. We attack a long-standing open question of Stanley that asks whether trees are distinguished by their CSFs. We prove new structural results about the CSFs of well-known subclasses of trees, including an involution on symmetric functions that swaps the CSFs of caterpillars in pairs, as well as a characterization of all linear relations between the CSFs of spiders. Then, we study a related graph invariant, the generalized degree polynomial (GDP), introduced by Crew and shown to be determined by the CSF by Aliste-Prieto et al. We present several classes of data about a tree that can be recovered from its GDP (and thus also from its CSF), such as the double-degree sequence and leaf adjacency sequence; this extends previous work of Martin, Morin, and Wagner. Then, we consider vector-valued and matrix-valued variants of the GDP for trees with distinguished vertices. By using linear relations that these GDPs satisfy, we prove new recurrence relations for the ordinary GDP and give several constructions that produce families of ordinary trees with the same GDP. In addition, we prove in some cases that the trees produced by such constructions have different CSFs. Our work suggests a program for further work using the GDP as an intermediary.
  
  
• (October 2025) Substring compatibility of permutation statistics.
  A permutation statistic is substring-compatible if its value on a permutation determines its value on every substring of that permutation. We construct the substring coalgebra of such a statistic, an analog of the shuffle algebra of a shuffle-compatible statistic introduced by Gessel and Zhuang. Furthermore, we show that for substring-compatible statistics that also satisfy a weak form of shuffle compatibility, the shuffle algebra and substring coalgebra can be combined to yield a Hopf algebra. Finally, we conjecture that the only nontrivial permutation statistics that are both shuffle-compatible and substring-compatible are the descent set, the peak set, and the valley set, and we describe our progress towards proving this conjecture.
  
  
• (November 2024) Generalized degree polynomials of trees (with Ricky Liu). To appear in Transactions of the American Mathematical Society.
  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). In Combinatorial Theory, Volume 6, Issue 1, 2026.
  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.