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• The sign character of the triagonal fermionic coinvariant ring. (2025). arXiv. Submitted.

  Abstract: We determine the trigraded multiplicity of the sign character of the triagonal fermionic coinvariant ring \(R_n^{(0,3)}\). As a corollary, this proves a conjecture of Bergeron (2020) that the dimension of the sign character of \(R_n^{(0,3)}\) is \(n^2-n+1\). We also give an explicit formula for double hook characters in the diagonal fermionic coinvariant ring \(R_n^{(0,2)}\), and discuss methods towards calculating the sign character of \(R_n^{(0,4)}\). Finally, we give a multigraded refinement of a conjecture of Bergeron (2020) that the dimension of the sign character of the \((1,3)\)-bosonic-fermionic coinvariant ring \(R_n^{(1,3)}\) is \(\frac{1}{2}F_{3n}\), where \(F_n\) is a Fibonacci number.

• Polyhedral geometry of refined \(q,t\)-Catalan numbers, with Matthias Beck, Mitsuki Hanada, Max Hlavacek, Andrés R. Vindas Meléndez, and Katie Waddle. (2024). arXiv. Accepted to Journal of Combinatorics.

  Abstract: We study a refinement of the \(q,t\)-Catalan numbers introduced by Xin and Zhang (2022, 2023) using tools from polyhedral geometry. These refined \(q,t\)-Catalan numbers depend on a vector of parameters \(\vec{k}\) and the classical \(q,t\)-Catalan numbers are recovered when \(\vec{k} = (1,\ldots,1)\). We interpret Xin and Zhang's generating functions by developing polyhedral cones arising from constraints on \(\vec{k}\)-Dyck paths and their associated area and bounce statistics. Through this polyhedral approach, we recover Xin and Zhang's theorem on \(q,t\)-symmetry of the refined \(q,t\)-Catalan numbers in the cases where \(\vec{k} = (k_1,k_2,k_3)\) and \((k,k,k,k)\), give some extensions, including the case \(\vec{k} = (k,k+m,k+m,k+m)\), and discuss relationships to other generalizations of the \(q,t\)-Catalan numbers.

• A conjectural basis for the \((1,2)\)-bosonic-fermionic coinvariant ring. (2024). arXiv. Submitted.

  Abstract: We give the first conjectural construction of a monomial basis for the coinvariant ring \(R_n^{(1,2)}\), for the symmetric group \(S_n\) acting on one set of bosonic (commuting) and two sets of fermionic (anticommuting) variables. Our construction interpolates between the modified Motzkin path basis for \(R_n^{(0,2)}\) of Kim-Rhoades (2022) and the super-Artin basis for \(R_n^{(1,1)}\) conjectured by Sagan-Swanson (2024) and proven by Angarone et al. (2024). We prove that our proposed basis has cardinality \(2^{n-1}n!\), aligning with a conjecture of Zabrocki (2020) on the dimension of \(R_n^{(1,2)}\), and show how it gives a combinatorial expression for the Hilbert series. We also conjecture a Frobenius series for \(R_n^{(1,2)}\). We show that these proposed Hilbert and Frobenius series are equivalent to conjectures of Iraci, Nadeau, and Vanden Wyngaerd (2023) on \(R_n^{(1,2)}\) in terms of segmented Smirnov words, by exhibiting a weight-preserving bijection between our proposed basis and their segmented permutations. We extend some of their results on the sign character to hook characters, and give a formula for the \(m_\mu\) coefficients of the conjectural Frobenius series. Finally, we conjecture a monomial basis for the analogous ring in type \(B_n\), and show that it has cardinality \(4^nn!\).

• The \(e\)-positivity of the chromatic symmetric function for twinned paths and cycles, with Esther Banaian, Kyle Celano, Megan Chang-Lee, Laura Colmenarejo, Owen Goff, Jamie Kimble, Lauren Kimpel, Jinting Liang, and Sheila Sundaram. (2024). arXiv. Submitted.

  Abstract: The operation of twinning a graph at a vertex was introduced by Foley, Hoàng, and Merkel (2019), who conjectured that twinning preserves \(e\)-positivity of the chromatic symmetric function. A counterexample to this conjecture was given by Li, Li, Wang, and Yang (2021). In this paper, we prove that \(e\)-positivity is preserved by the twinning operation on cycles, by giving an \(e\)-positive generating function for the chromatic symmetric function, as well as an \(e\)-positive recurrence. We derive similar \(e\)-positive generating functions and recurrences for twins of paths. Our methods make use of the important triple deletion formulas of Orellana and Scott (2014), as well as new symmetric function identities.

• Generalized parking function polytopes, with Mitsuki Hanada and Andrés R. Vindas Meléndez. Annals of Combinatorics, 28, 575–613, (2024). arXiv.

  Abstract: A classical parking function of length \(n\) is a list of positive integers \((a_1, a_2, \ldots, a_n)\) whose nondecreasing rearrangement \(b_1 \leq b_2 \leq \cdots \leq b_n\) satisfies \(b_i \leq i\). The convex hull of all parking functions of length \(n\) is an \(n\)-dimensional polytope in \(\mathbb{R}^n\), which we refer to as the classical parking function polytope. Its geometric properties have been explored in (Amanbayeva and Wang 2022) in response to a question posed in (Stanley 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of \(\mathbf{x}\)-parking functions for \(\mathbf{x}=(a,b,\dots,b)\), which refer to as \(\mathbf{x}\)-parking function polytopes. We explore connections between these \(\mathbf{x}\)-parking function polytopes, the Pitman-Stanley polytope, and the partial permutahedra of (Heuer and Striker 2022). In particular, we establish a closed-form expression for the volume of \(\mathbf{x}\)-parking function polytopes. This allows us to answer a conjecture of (Behrend et al. 2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary.

Spring 2025: I am co-organizing the Combinatorics research seminar with Christian Gaetz, Nicolle González, and Mitsuki Hanada.

Fall 2024: I am co-organizing the Combinatorics research seminar with Christian Gaetz, Nicolle González, and Mitsuki Hanada. I am co-organizing the Fall 2024 BAD Math Day on December 7 at UC Berkeley.

Summer 2024: I am co-organizing the 21st Fibonacci Conference from July 8-12 at Harvey Mudd College in Claremont, CA.

Spring 2024: I am co-organizing the Combinatorics research seminar and the Combinatorics student seminar with Nicolle González and Mitsuki Hanada.

Fall 2023: I am co-organizing the Combinatorics research seminar with Nicolle González and Mitsuki Hanada.

Spring 2023: I am co-organizing the Combinatorics research seminar with Andrés R. Vindas Meléndez, Christopher Ryba, and Mitsuki Hanada.

Fall 2022: I am co-organizing the Combinatorics student seminar with Mitsuki Hanada. I am also co-organizing the Combinatorics research seminar with Andrés R. Vindas Meléndez, Christopher Ryba, and Mitsuki Hanada.

USC Combinatorics Seminar; The \((1,2)\)-bosonic-fermionic coinvariant ring; October 7, 2024; Los Angeles, CA

UCLA Combinatorics Seminar; The \((1,2)\)-bosonic-fermionic coinvariant ring; October 4, 2024; Los Angeles, CA

UC Berkeley Combinatorics Seimnar; The \((1,2)\)-bosonic-fermionic coinvariant ring; September 25, 2024; Berkeley, CA

Graduate Onlince Combinatorics Colloquium; The \((1,2)\)-bosonic-fermionic coinvariant ring; September 18, 2024; Online

AMS Western Sectional at SFSU; Special session on research in combinatorics by early career mathematicians; \(q,t\)-Catalan Objects; May 4, 2024; San Francisco, CA

Graduate Student Combinatorics Conference at WUSTL; The volume of a generalized parking function polytope; March 18, 2023; St. Louis, MO

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• A tiling interpretation of a generalized Zeckendorf's theorem, with Arthur T. Benjamin. Integers 22, Paper A104, 35 p. (2022).

  Abstract: Zeckendorf's theorem states that every positive integer can be decomposed uniquely into a sum of non-consecutive Fibonacci numbers (where \(f_1 = 1\) and \(f_2 = 2\)). Previous work by Grabner and Tichy (1990) and Miller and Wang (2012) has found a generalization of Zeckendorf's theorem to a larger class of recurrent sequences, called Positive Linear Recurrence Sequences (PLRS's). We apply well-known tiling interpretations of recurrence sequences from Benjamin and Quinn (2003) to PLRS's. We exploit that tiling interpretation to create a new tiling interpretation specific to PLRS's that captures the behavior of the generalized Zeckendorf's theorem.

• An introduction to completeness of positive linear recurrence sequences, with Elżbieta Bołdyriew, John Haviland, Phúc Lâm, Steven J. Miller, and Fernando Trejos Suárez. Fibonacci Q. 58, No. 5, 77-90 (2020).

  Abstract: A positive linear recurrence sequence (PLRS) is a sequence defined by a homogeneous linear recurrence relation with positive coefficients and a particular set of initial conditions. A sequence of positive integers is \emph{complete} if every positive integer is a sum of distinct terms of the sequence. One consequence of Zeckendorf's theorem is that the sequence of Fibonacci numbers is complete. Previous work has established a generalized Zeckendorf's theorem for all PLRS's. We consider PLRS's and want to classify them as complete or not. We study how completeness is affected by modifying the recurrence coefficients of a PLRS. Then, we determine in many cases which sequences generated by coefficients of the forms \([1, \ldots, 1, 0, \ldots, 0, N]\) are complete. Further, we conjecture bounds for other maximal last coefficients in complete sequences in other families of PLRS's. Our primary method is applying Brown's criterion, which says that an increasing sequence \(\{H_n\}_{n = 1}^{\infty}\) is complete if and only if \(H_1 = 1\) and \(H_{n + 1} \leq 1 + \sum_{i = 1}^n H_i\). This paper is an introduction to the topic that is explored further in arXiv:2010.01655[math.CO].

• Counting on Euler and Bernoulli number identities, with Arthur T. Benjamin and Thomas C. Martinez. Fibonacci Q. 58, No. 5, 30-33 (2020).

  Abstract: While there are many identities involving the Euler and Bernoulli numbers, they are usually proved analytically or inductively. We prove two identities involving Euler and Bernoulli numbers with combinatorial reasoning via up-down permutations.

Graphical Models in Algebraic Combinatorics at SLMath; June 23-July 3, 2025; Berkeley, CA

AMS Western Sectional at Cal Poly SLO; May 3-4, 2025; San Luis Obispo, CA

Latinx in the Mathematical Sciences at IPAM; Poster; March 6-8, 2025; Los Angeles, CA

Algebra, Combinatorics, and Discrete Geometry at IPAM; Poster; February 10-14, 2025; Los Angeles, CA

Bay Area Discrete Math Day; December 7, 2024; Berkeley, CA

Formal Power Series and Algebraic Combinatorics; July 22-26, 2024; Bochum, Germany

Summer School in Algebraic Combinatorics at MPI Leipzig; July 15-19, 2024; Leipzig, Germany

Fibonacci Conference at HMC; July 8-12, 2024; Claremont, CA

Combinatorial Legacies of Richard P. Stanley at Harvard; June 3-7, 2024; Cambridge, MA

AMS Western Sectional at SFSU; Speaker; May 4-5, 2024; San Francisco, CA

Georgia Benkart Conference at SLMath; May 1-3, 2024; Berkeley, CA

Bay Area Discrete Math Day; April 20, 2024; Santa Clara, CA

Integrability and Algebraic Combinatorics at IPAM; April 15-19, 2024; Los Angeles, CA

Combinatorial Algebra meets Algebraic Combinatorics at LACIM; January 26-28, 2024; Montréal, QC

Joint Mathematics Meetings; January 3-6, 2024; San Francisco, CA

Dimers: Combinatorics, Representation Theory and Physics at the Graduate Center, CUNY; August 14-25, 2023; New York, NY

Graduate Research Workshop in Combinatorics; July 23-August 4, 2023; Laramie, WY

Formal Power Series and Algebraic Combinatorics; July 17-21, 2023; Davis, CA

Lake Michigan Workshop on Combinatorics and Graph Theory; May 13-14, 2023; South Bend, IN

AMS Western Sectional at CSU Fresno; May 6-7, 2023; Fresno, CA

AIM Workshop - Gems of Combinatorics; March 27-31, 2023; San José, CA

Graduate Student Combinatorics Conference at WUSTL; Speaker; March 17-19, 2023; St. Louis, MO

Latinx in the Mathematical Sciences at IPAM; Math Circle co-lead; July 7-9, 2022; Los Angeles, CA

Spring 2025: GSI for Math 104: Introduction to Analysis.

Fall 2024: GSI for Math 55: Discrete Mathematics (Sections 116 & 117) with Nikhil Srivastava. GSI for Math 16B: Analytic Geometry and Calculus (Sections 120 & 124) with Ryan Hass.

Fall 2023: Head GSI for Math 55: Discrete Mathematics (Section 113) with Sylvie Corteel.

Summer 2022: Instructor for Math N55: Discrete Mathematics (Section 003).

Spring 2022: GSI for Math 1B: Calculus (Sections 203 & 207) with Sung-Jin Oh.