Research

My research is in algebraic combinatorics. A central problem in algebraic combinatorics is to show that certain symmetric functions expand with positive integer coefficients in certain bases. A positive integer coefficient could count the number of some combinatorial objects. It could also be an intersection number of varieties, a Betti number of a topological space, or a rank of a module, connecting combinatorics to geometry, topology, and representation theory. During my Ph.D. at UC Berkeley, I studied Schur-positivity of LLT symmetric functions. During my position at MIT, I studied e-positivity of chromatic (quasi)-symmetric functions. I am continuing this work at Dartmouth College.

Submitted

• The chromatic symmetric function of graphs glued at a single vertex with Aarush Vailaya
  

  We describe how the chromatic symmetric function of two graphs glued at a single vertex can be expressed as a matrix multiplication using certain information of the two individual graphs. We then prove new e-positivity results by using a connection between forest triples, defined by the first author, and Hikita's probabilities associated to standard Young tableaux. Specifically, we prove that gluing a sequence of unit interval graphs and cycles results in an e-positive graph. We also prove e-positivity for a graph obtained by gluing the first and last vertices of such a sequence. This generalizes e-positivity of cycle-chord graphs and supports Ellzey's conjectured e-positivity for proper circular arc digraphs.

• Chromatic symmetric functions and change of basis with Bruce Sagan
  

  We prove necessary conditions for certain elementary symmetric functions, e_\lambda, to appear with nonzero coefficient in Stanley's chromatic symmetric function as well as in the generalization considered by Shareshian and Wachs. We do this by first considering the expansion in the monomial or Schur basis and then performing a basis change. Using the former, we make a connection with two fundamental graph theory invariants, the independence and clique numbers. This allows us to prove nonnegativity of three-column coefficients for all natural unit interval graphs. The Schur basis permits us to give a new interpretation of the coefficient of e_n in terms of tableaux. We are also able to give an explicit formula for that coefficient.

Accepted

• Graphs missing a connected partition
  Advances in Applied Mathematics (2025)

  We prove that a graph with a cut vertex whose deletion produces at least five connected components must be missing a connected partition of some type. We prove that this also holds if there are four connected components that each have at least two vertices. In particular, the chromatic symmetric function of such a graph cannot be $e$-positive. This brings us very close to the conjecture by Dahlberg, She, and van Willigenburg of non-$e$-positivity for all trees with a vertex of degree at least four. We also prove that spiders with four legs cannot have an $e$-positive chromatic symmetric function.

• Adjacent cycle chains are e-positive with Aarush Vailaya
  Electronic Journal of Combinatorics (2025)

  We describe a way to decompose the chromatic symmetric function as a positive sum of smaller pieces. We show that these pieces are e-positive for cycles. Then we prove that attaching a cycle to a graph preserves the e-positivity of these pieces. From this, we prove an e-positive formula for graphs of cycles connected at adjacent vertices. We extend these results to graphs formed by connecting a sequence of cycles and cliques.

Appeared

• A signed e-expansion of the chromatic quasisymmetric function
  Combinatorial Theory, Vol. 5 No. 2 (2025)

  We prove a new signed elementary symmetric function expansion of the chromatic quasisymmetric function of any natural unit interval graph. We then use a sign-reversing involution to prove a new combinatorial formula for K-chains, which are graphs formed by joining cliques at single vertices. This formula immediately implies e-positivity and e-unimodality for K-chains. We also prove a version of our signed e-expansion for arbitrary graphs. We prove a new signed elementary symmetric function expansion of the chromatic quasisymmetric function of any natural unit interval graph. We then use a sign-reversing involution to prove a new combinatorial formula for K-chains, which are graphs formed by joining cliques at single vertices. This formula immediately implies e-positivity and e-unimodality for K-chains. We also prove a version of our signed e-expansion for arbitrary graphs.

• A horizontal-strip LLT polynomial is determined by its weighted graph
  Combinatorial Theory, Vol. 3 No. 3 (2023)

  We prove that two horizontal-strip LLT polynomials are equal if the associated weighted graphs defined by the author in a previous paper are isomorphic. This provides a sufficient condition for equality of horizontal-strip LLT polynomials and yields a well-defined LLT polynomial indexed by a weighted graph. We use this to prove some new relations between LLT polynomials and we explore a connection with extended chromatic symmetric functions.

• A combinatorial Schur expansion of triangle-free horizontal-strip LLT polynomials
  Combinatorial Theory, Vol. 1 No. 14 (2021)

  In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial G_\lambda(x;q) in some special cases. We associate a weighted graph \Pi to \lambda and we use it to express a linear relation among LLT polynomials. We apply this relation to prove an explicit combinatorial Schur-positive expansion of G_\lambda(x;q) whenever \Pi is triangle-free. We also prove that the largest power of q in the LLT polynomial is the total edge weight of our graph.

• Classifying the near-equality of ribbon Schur functions
  European Journal of Combinatorics, Vol. 90, 103197 (2020)

  We consider the problem of determining when the difference of two ribbon Schur functions is a single Schur function. We fully classify the five infinite families of pairs of ribbon Schur functions whose difference is a single Schur function with corresponding partition having at most two parts at least 2. We also prove an identity for differences of ribbon Schur functions and we determine some necessary conditions for such a difference to be Schur-positive, depending on the distribution of 1's and the end row lengths. We consider the problem of determining when the difference of two ribbon Schur functions is a single Schur function. We fully classify the five infinite families of pairs of ribbon Schur functions whose difference is a single Schur function with corresponding partition having at most two parts at least 2. We also prove an identity for differences of ribbon Schur functions and we determine some necessary conditions for such a difference to be Schur-positive, depending on the distribution of 1's and the end row lengths.

• The number of rational points of hyperelliptic curves over subsets of finite fields with Kristina Nelson, Jozsef Solymosi, and Ching Wong
  Involve, a Journal of Mathematics, Vol. 12 No. 5, 755--765 (2019)

  We prove two related concentration inequalities concerning the number of rational points of hyperelliptic curves over subsets of a finite field. In particular, we investigate the probability of a large discrepancy between the numbers of quadratic residues and non-residues in the image of such subsets over uniformly random hyperelliptic curves of given degrees. We find a constant probability of such a high difference and show the existence of sets with an exceptionally large discrepancy.

• Necessary conditions for Schur-maximality with Stephanie van Willigenburg
  Electronic Journal of Combinatorics, Vol. 25, P2.30 (2018)

  McNamara and Pylyavskyy conjectured precisely which connected skew shapes are maximal in the Schur-positivity order, which says that B\leq _s A if s_A-s_B is Schur-positive. Towards this, McNamara and van Willigenburg proved that it suffices to study equitable ribbons, namely ribbons whose row lengths are all of length a or (a+1) for a\geq 2. In this paper we confirm the conjecture of McNamara and Pylyavskyy in all cases where the comparable equitable ribbons form a chain. We also confirm a conjecture of McNamara and van Willigenburg regarding which equitable ribbons in general are minimal. Additionally, we establish two sufficient conditions for the difference of two ribbons to be Schur-positive, which manifest as diagrammatic operations on ribbons. We also deduce two necessary conditions for the difference of two equitable ribbons to be Schur-positive that rely on rows of length a being at the end, or on rows of length (a+1) being evenly distributed.

Refereed Conference

• On e-positivity of trees and connected partitions
  

• A signed e-expansion of the chromatic symmetric function and some new e-positive graphs
  Seminaire Loharingien de Combinatoire 91B.48 (2024)

• Horizontal-strip LLT polynomials
  Seminaire Loharingien de Combinatoire 89B.40 (2023)

• A combinatorial Schur expansion of triangle-free horizontal-strip LLT polynomials
  Seminaire Loharingien de Combinatoire 85B.18 (2021)

• Schur-positivity of equitable ribbons with Stephanie van Willigenburg
  Seminaire Loharingien de Combinatoire 80B.18 (2018)