Current projects include:

• Arithmetical Structures on Graphs: Graphs are collections of vertices, some of which are connected by edges. Arithmetical structures are integer labels that we place on the vertices of the graph that satisfy some divisibility condition. In this project, students will have the opportunity to explore different hands-on problems related to these labelings.

• Metrics on Sets of Permutations: A permutation written in one-line notation has a peak at position j if the value in the j-th position is greater than the value of its immediate neighbors. A permutation has a descent at position j if the value in the j-th position is larger than the value immediately to the right.  The set of all permutations can be partitioned based on the locations of peaks or descents in each permutation. At the same time, a variety of different notions of the distance between permutations are used in areas including rankings, statistics, coding theory, and other applications. This project will focus on studying metric properties of sets of permutations with similar peaks and/or descents.

• Tower of Hanoi: The Tower of Hanoi is an old wooden puzzle consisting of three rods and a given collection of ordered disks. The goal of the puzzle is to move all disks from one rod to another following some simple rules. For this puzzle, there is a unique efficient solution and it is usually described in a recursive manner. The main goal of this project is to produce a non-recursive description of the exact distribution of the disks at given step m when performing the most efficient solution of the puzzle.

[16] A. Diaz-Lopez, K. Haymaker, C. McGarry, D. McMahon. Metrics on permutations with the same descent set. (Submitted). arXiv version

[15] R. Behrend, F. Castillo, A. Chavez, A. Diaz-Lopez, L. Escobar, P. E. Harris, and E. Insko. Partial Permutohedra. (Submitted.) arXiv version

[14] A. Diaz-Lopez, K. Haymaker, and M. Tait. Spectral Radii of Arithmetical Structures on Cycle Graphs. (Submitted.) arXiv version

[13] A. Diaz-Lopez, K. Haymaker, K. Keough, J. Park, and E. White. Metrics on permutations with the same peak set. (To appear in Involve),  arXiv version

[12] K. Archer, A. Diaz-Lopez, D. Glass, and J. Louwsma. Critical groups of arithmetical structures on star graphs and complete graphs.  The Electronic Journal of Combinatorics 31 (1) (2024), #P1.5. arXiv version

[11] A. Diaz-Lopez and J. Louwsma. Critical groups of arithmetical structures under a generalized star-clique operation. Linear Algebra and its Applications 656 (2023), pp. 324-344. arXiv version

[10] A. Diaz-Lopez, P. Harris, I. Huang, E. Insko, and L. Nilsen. A formula for enumerating permutations with a fixed pinnacle set. Discrete Math. 344 (2021), no. 6, 112375. arXiv version

[9] (Student author: A. Vetter*, Advisor: A. Diaz-Lopez) Arithmetical Structures on E_n graphs. Villanova Veritas Journal. Vol. 3 No. 1 (2021).

[8] K. Archer, A. Bishop, A. Diaz-Lopez, L. Garcia-Puente, D. Glass, and J. Louwsma. Arithmetical Structures on Bidents. Discrete Math. 343 (2020), no. 7, 111850, 23 pp. 05C50 (05A10). arXiv version.

[7] A. Diaz-Lopez, L. Everham, P. Harris, E. Insko, V. Marcantonio, and M. Omar. Counting peaks on graphs. Australas. J. Combin. 75 (2019), 174–189. 05A05 (05A10 05A15). arXiv version.

[6] A. Diaz-Lopez, P. Harris, E. Insko, M. Omar, and B. Sagan. Descent Polynomials. Discrete Math Vol. 342 (2019), Num. 6, pp 1674 - 1686. arXiv version.

[5] A. Diaz-Lopez P. Harris, E. Insko, and M. Omar. A proof of the peak polynomial positivity conjecture.  Sém. Lothar. Combin. 78B (2017), Art. 6, 9 pp.

[4] A. Diaz-Lopez, P. Harris, E. Insko, and M. Omar. A proof of the peak polynomial positivity conjecture. Journal of Combinatorial Theory, Series A 149C (2017) pp. 21-29. arXiv version.

[3] F. Castro-Velez, A. Diaz-Lopez, R. Orellana, J. Pastrana, and R. Zevallos. Number of permutations with the same peak set for signed permutations. J. Comb. 8 (2017), no. 4, 631-652. arXiv version.

[2] A. Diaz-Lopez, P. Harris, E. Insko, and D. Perez-Lavin. Peaks sets of classical Coxeter groups. Involve 10-2 (2017), 263--290. DOI 10.2140/involve.2017.10.263. arXiv version.

[1] A. Diaz, N. Harman, S.Howe, and D. Thompson. Isoperimetric Problems in Sectors with Density. Adv. Geom. 12 (2012), 589-619. arXiv version.

• 2022-2024 NSF LEAPS Grant: Combinatorics from an Algebraic and Geometric Lens, $249,937.

• 2021 Villanova University Summer Research Grant

• 2019 Villanova University Summer Research Grant

• 2018 CURM Mini-grant, $25,000

• 2018 IAS Summer Collaborator Grant

The following Villanova students have been part of the Diaz-Lopez Research Group.