Research
My research interests lie primarily in number theory, combinatorics, and algebra, with a particular interest in discrete structures, recurrence sequences, and computational approaches to mathematical discovery.
A recurring theme in my work is the study of Fibonacci-type sequences and the combinatorial structures associated with them. I have worked on questions involving greatest common divisors, divisibility, generalized Hosoya triangles, lattice paths, and non-decreasing Dyck paths. I am also interested in oriented matroids, algebraic structures, and the use of computation to explore patterns, test conjectures, and guide the development of new results.
Research Areas
Number Theory & Recurrence Sequences
Fibonacci and generalized Fibonacci sequences, divisibility, greatest-common-divisor identities, Hosoya-type structures, and related arithmetic questions.
Combinatorics
Non-decreasing Dyck paths, lattice paths, enumerative combinatorics, generating functions, and discrete structures.
Algebra & Oriented Matroids
Oriented matroids, algebraic structures, duality, automorphism groups, and related combinatorial-algebraic questions.
Computational Mathematics
The use of computation to investigate examples, test conjectures, discover patterns, and support theoretical mathematical research.
Current Research
Generalized Fibonacci GCD Structures
Current work on exact greatest-common-divisor formulas, arithmetic progressions of indices, generalized Fibonacci sequences, and the structural behavior of GCDs across families of recurrence terms.
Automorphism Groups
Research concerning automorphism groups of algebraic constructions, including structural conditions under which automorphism groups become nonabelian.
Aperiodic Tilings and Graphs
Research on graph-theoretic structures associated with aperiodic tilings, including local vertex configurations and combinatorial structures arising from Ammann tilings.
Computational & Experimental Mathematics
The use of Python, SageMath, GAP, and related computational tools to explore mathematical examples, identify patterns, test conjectures, and guide theoretical work.
Selected Publications
- 2025 Counting Subwords in Non-Decreasing Dyck Paths Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.6 · Abstract
- 2019 Enumerating Several Aspects of Non-Decreasing Dyck Paths Discrete Mathematics, 342.10 (2019), 3079–3097 · DOI
- 2015 Some Enumerations on Non-Decreasing Dyck Paths Electronic Journal of Combinatorics, Vol. 22, Issue 1 (2015) · DOI
- 2014 GCD Property of the Generalized Star of David in the Generalized Hosoya Triangle Journal of Integer Sequences, Vol. 17 (2014), Article 14.3.6 · Abstract
- 2014 9-Modularity and GCD Properties of Generalized Fibonacci Numbers Electronic Journal of Combinatorial Number Theory, Vol. 14 (2014), Article A55 · PDF
- 2012 GCD Properties in Hosoya's Triangle The Fibonacci Quarterly, 50.2 (2012), 163–174 · PDF
Undergraduate Research
Undergraduate research is an important part of my work. I have founded and directed research groups involving students in the United States and Colombia. These collaborations have led to dozens of peer-reviewed publications, conference presentations, research awards, and other opportunities for students to participate directly in mathematical research.
I view undergraduate research as an opportunity for students to experience mathematics not simply as a collection of established results, but as an active process of exploration, conjecture, problem solving, and discovery.