Guilherme Zeus Dantas e Moura
Publications
Research Articles
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Probabilistic Entry Swapping Bijections for Non-Attacking Fillings (2025)
Abstract
Non-attacking fillings are combinatorial objects central to the theory of Macdonald polynomials. A probabilistic bijection for partition-shaped non-attacking fillings was introduced by Mandelshtam (2024) to prove a compact formula for symmetric Macdonald polynomials. In this work, we generalize this probabilistic bijection to composition-shaped non-attacking fillings. As an application, we provide a bijective proof to extend a symmetry theorem for permuted-basement Macdonald polynomials established by Alexandersson (2019), proving a version with fewer assumptions.
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Towards the Gaussianity of random Zeckendorf games (2025)
Combinatorial and Additive Number Theory VI. CANT, New York, USA, 2022 and 2023, pp. 65–99. Springer Proceedings in Mathematics & Statistics, 464.Abstract
Zeckendorf proved that any positive integer has a unique decomposition as a sum of non-consecutive Fibonacci numbers, indexed by $F_1 = 1, F_2 = 2, F_{n+1} = F_n + F_{n-1}$. Motivated by this result, Baird, Epstein, Flint, and Miller defined the two-player Zeckendorf game, where two players take turns acting on a multiset of Fibonacci numbers that always sums to $N$. The game terminates when no possible moves remain, and the final player to perform a move wins. Notably, studied the setting of random games: the game proceeds by choosing an available move uniformly at random, and they conjecture that as the input $N \to \infty$, the distribution of random game lengths converges to a Gaussian. We prove that certain sums of move counts is constant, and find a lower bound on the number of shortest games on input $N$ involving the Catalan numbers. The works Baird et al. and Cuzensa et al. determined how to achieve a shortest and longest possible Zeckendorf game on a given input $N$, respectively: we establish that for any input $N$, the range of possible game lengths constitutes an interval of natural numbers: every game length between the shortest and longest game lengths can be achieved. We further the study of probabilistic aspects of random Zeckendorf games. We study two probability measures on the space of all Zeckendorf games on input $N$: the uniform measure, and the measure induced by choosing moves uniformly at random at any given position. Under both measures that in the limit $N \to \infty$, both players win with probability $1/2$. We also find natural partitions of the collection of all Zeckendorf games of a fixed input $N$, on which we observe weak convergence to a Gaussian in the limit $N \to \infty$. We conclude the work with many open problems.
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Cluster monomials in graph Laurent phenomenon algebras (2024)
To appear in Algebraic CombinatoricsAbstract
Laurent Phenomenon algebras, first introduced by Lam and Pylyavskyy, are a generalization of cluster algebras that still possess many salient features of cluster algebras. Graph Laurent Phenomenon algebras, defined by Lam and Pylyavskyy, are a subclass of Laurent Phenomenon algebras whose structure is given by the data of a directed graph. The main result of this paper is that the cluster monomials of a graph Laurent Phenomenon algebra form a linear basis, conjectured by Lam and Pylyavskyy and analogous to a result for cluster algebras by Caldero and Keller.
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On generic universal rigidity on the line (2023)
Proceedings of the 12th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications, Budapest, March 2023, pp. 219–228Abstract
A $d$-dimensional bar-and-joint framework $(G,p)$ with underlying graph $G$ is called universally rigid if all realizations of $G$ with the same edge lengths, in all dimensions, are congruent to $(G,p)$. A graph $G$ is said to be generically universally rigid in $\mathbb{R}^d$ if every $d$-dimensional generic framework $(G,p)$ is universally rigid. In this paper we focus on the case $d=1$. We give counterexamples to a conjectured characterization of generically universally rigid graphs from R. Connelly (2011). We also introduce two new operations that preserve the universal rigidity of generic frameworks, and the property of being not universally rigid, respectively. One of these operations is used in the analysis of one of our examples, while the other operation is applied to obtain a lower bound on the size of generically universally rigid graphs. This bound gives a partial answer to a question from T. Jordán and V-H. Nguyen (2015).
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Fibonacci Quarterly, 61(3), pp. 257–274Abstract
Zeckendorf's Theorem implies that the Fibonacci number $F_n$ is the smallest positive integer that cannot be written as a sum of non-consecutive previous Fibonacci numbers. Catral et al. studied a variation of the Fibonacci sequence, the Fibonacci Quilt sequence: the plane is tiled using the Fibonacci spiral, and integers are assigned to the squares of the spiral such that each square contains the smallest positive integer that cannot be expressed as the sum of non-adjacent previous terms. This adjacency is essentially captured in the differences of the indices of each square: the $i$-th and $j$-th squares are adjacent if and only if $|i - j| \in \{1, 3, 4\}$ or $\{i, j\} = \{1, 3\}$. We consider a generalization of this construction: given a set of positive integers $S$, the $S$-legal index difference ($S$-LID) sequence $(a_n)_{n=1}^\infty$ is defined by letting $a_n$ to be the smallest positive integer that cannot be written as $\sum_{\ell \in L} a_\ell$ for some set $L \subset [n-1]$ with $|i - j| \notin S$ for all $i, j \in L$. We discuss our results governing the growth of $S$-LID sequences, as well as results proving that many families of sets $S$ yield $S$-LID sequences which follow simple recurrence relations.
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Outerplanar Turán number of a cycle (2023)
Abstract
A graph is outerplanar if it has a planar drawing for which all vertices belong to the outer face of the drawing. Let $H$ be a graph. The outerplanar Turán number of $H$, denoted by $ex_\mathcal{OP}(n,H)$, is the maximum number of edges in an $n$-vertex outerplanar graph which does not contain $H$ as a subgraph. In 2021, L. Fang et al. determined the outerplanar Turán number of cycles and paths. In this paper, we use techniques of dual graph to give a shorter proof for the sharp upperbound of $ex_\mathcal{OP}(n,C_k)\leq \frac{(2k - 5)(kn - k - 1)}{k^2 - 2k - 1}$.
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Sum and difference sets in generalized dihedral groups (2022)
Abstract
Given a group $G$, we say that a set $A \subseteq G$ has more sums than differences (MSTD) if $|A+A| > |A-A|$, has more differences than sums (MDTS) if $|A+A| < |A-A|$, or is sum-difference balanced if $|A+A| = |A-A|$. A problem of recent interest has been to understand the frequencies of these type of subsets. The seventh author and Vissuet studied the problem for arbitrary finite groups $G$ and proved that almost all subsets $A\subseteq G$ are sum-difference balanced as $|G|\to\infty$. For the dihedral group $D_{2n}$, they conjectured that of the remaining sets, most are MSTD, i.e., there are more MSTD sets than MDTS sets. Some progress on this conjecture was made by Haviland et al. in 2020, when they introduced the idea of partitioning the subsets by size: if, for each $m$, there are more MSTD subsets of $D_{2n}$ of size $m$ than MDTS subsets of size $m$, then the conjecture follows. We extend the conjecture to generalized dihedral groups $D=\mathbb{Z}_2\ltimes G$, where $G$ is an abelian group of size $n$ and the nonidentity element of $\mathbb{Z}_2$ acts by inversion. We make further progress on the conjecture by considering subsets with a fixed number of rotations and reflections. By bounding the expected number of overlapping sums, we show that the collection $\mathcal S_{D,m}$ of subsets of the generalized dihedral group $D$ of size $m$ has more MSTD sets than MDTS sets when $6\le m\le c_j\sqrt{n}$ for $c_j=1.3229/\sqrt{111+5j}$, where $j$ is the number of elements in $G$ with order at most $2$. We also analyze the expectation for $|A+A|$ and $|A-A|$ for $A\subseteq D_{2n}$, proving an explicit formula for $|A-A|$ when $n$ is prime.
Other Manuscripts
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An invitation to Fibonacci digits (2024)
To appear in the Fibonacci QuarterlyAbstract
The purpose of this short note is to show the interplay between math outreach and conducting original research, in particular how each can build off the other.
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Combinatorial Models for Key Polynomials and Atom Polynomials (2024)
Bachelor's thesis for a major in Mathematics at Haverford CollegeAbstract
We present a survey on some combinatorial models for key polynomials and atom polynomials, in the search for a combinatorial model for the coefficients of the product of two key polynomials in either the key polynomial basis or the atom polynomial basis.
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Requirement of Variable-Introducing Elements on Event Quantification in Chinese (2023)
Bachelor's thesis for a major in Linguistics at Haverford CollegeAbstract
Chinese uses the 每–都 měi–dōu cooccurrence to express universal quantification. The case where 每 měi is followed by a noun phrase (měi–NP) is well studied, but the case where 每 měi is followed by a verb phrase (měi–VP) is often forgotten in the literature. Huang (1996) proposed a hypothesis about 每 měi–vp constructions in Chinese: the requirement of a variable-introducing element in the verb phrase. Common variable-introducing elements that are used in 每 měi–VP sentences include 一次 yī cì ‘one time’ and 一场 yī chǎng ‘one event’, which in some sense quantify over the eventuality denoted by the verb phrase. However, Huang (1996) shows an example where an indefinite direct object, namely 一篇文章 yī piān wénzhāng ‘one article’, a variable-introducing element, also serves this role. This thesis strengthens the hypothesis of Huang (1996) by showing that it correctly predicts the grammaticality of a larger collection of sentences. The larger dataset includes ditransitive verbs, the 把 bǎ construction, and the verb copying construction, as well as includes sentences with two kinds of variable-introducing elements: indefinite expressions and the reflexive pronoun 自己 zìjǐ ‘self’.