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

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.

Please click on the paper title to show or hide the abstract.

• Generalized parking function polytopes, with Mitsuki Hanada and Andrés R. Vindas Meléndez. arXiv:2212.06885 [math.CO] (2022).

  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 the first closed-form expression for the volume of the convex hull of classical parking functions as a corollary.

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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.

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

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

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

• Summer 2022: Instructor for Math N55: Discrete Mathematics (Section 003).
• Spring 2022: GSI for Math 1B: Calculus (Sections 203 & 207) with Sung-Jin Oh.