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(2025) Together with Diego Villamizar, we investigated colored random tilings on grids, a combinatorial structure arising from a Mathematics Stack Exchange problem. We extended the study to general m×n grids with k colors, analyzing the number of tiling configurations using generating functions. Our results include explicit formulas for the expected value and variance of the number of polyominoes, with particular cases solved for m=1,2,3. Additionally, we explored hexagonal grids and formulated a recurrence relation for the number of distinct last columns in a tiling. Monte Carlo simulations provided further insights into expected tiling behavior. These findings are detailed in our paper Colored Random Tilings on Grids, published in Journal of Automata, Languages and Combinatorics.
(2025) Together with Jean-Luc, Sergey, and Diego, we investigated Motzkin polyominoes, a combinatorial structure derived from Motzkin words. These polyominoes are represented as bargraphs with columns determined by the values in Motzkin words. Our study reveals bijections between Motzkin polyominoes, restricted Catalan words, and primitive Łukasiewicz paths. Using generating functions, we analyzed statistics such as area, semiperimeter, and interior points, deriving closed-form expressions and asymptotic formulas. These findings are detailed in our paper The Combinatorics of Motzkin Polyominoes, (in Discrete Applied Mathematics).
See this presentation of Diego in ICECA2024.
(11/2024) Together with Jean-Luc, Nathanael, and Sergey, we enumerate zigzag knight’s paths—a variant of lattice paths inspired by the moves of a chess knight—under specific constraints. Our study examines properties such as height, altitude, and size, employing generating functions and asymptotic methods to analyze scenarios such as confinement between horizontal lines or within tubes. Additionally, we establish bijections between these paths and integer compositions. The results include closed-form expressions, bijections, and approximations, which are detailed in our paper Grand zigzag knight's paths, (in Enumerative Combinatorics and Applications).
See my presentation... PDF
(09/2024) with Jean-Luc, we enumerate lattice path contained in walls. We prove algebraically that paths in walls of a given width and ending on the x-axis are enumerated by the Catalan numbers, and we provide a bijection between these paths and Dyck paths. We also find that paths in walls with a given number of steps are enumerated by the Fibonacci numbers! Finally, we give a constructive bijection between the paths in walls of a given length and peakless Motzkin paths of the same length. Our results have been published in the paper titled Fibonacci and Catalan paths in a Wall, (in Discrete Mathematics).
(08/2024) With Beáta Benyi and Toufik Mansour, we study pattern avoidance in weak ascent sequences. In particular, we obtain the enumeration for several patterns of length 3. This is an analogous study to one given by Duncan and Steingrimsson (2011) for ascent sequences. Our results have been published in the paper titled Pattern Avoidance in Weak Ascent Sequences (in Discrete Mathematics & Theoretical Computer Science).
(07/2024) With Rigo and Diego, we explored a new variations of bargraphs (polyominoes) called non-decreasing bargraphs (inspired by the buildings on the campus of The Citadel). We establish intriguing connections between these novel objects and unimodal compositions. Our results have been published in the paper titled Restricted bargraphs and unimodal compositions (in Journal of Combinatorial Theory, Series A).
(07/2024) With Rigo, we introduced both symmetric and asymmetric statistic in Dyck paths. Now, in collaboration with Rigo and Jean-Luc, we have investigated this interesting parameter in Motzkin paths with air pocktes. Our results have been published in the paper titled Counting symmetric and asymmetric peaks in Motzkin paths with air pockets (in The Australasian Journal of Combinatorics).
(05/2004) Jonly with Jean-Luc Baril and Fabio Velandia found a bijection between the set of directed column-convex polyominoes on triangular and honeycomb lattices of area n and some families of restricted compositions. This is an analogous result to one given by Deutsch and Prodinger for polyominoes over square lattices. As a byproduct, we deduce new close forms for the number of hexagonal and triangular directed column-convex polyominoes of area n with k columns. (See our paper in Theoretical Computer Sciences)
(04/2024) During the 20th International Conference on Fibonacci Numbers and Their Applications (2022) , Brian Hopkins talked about Arndt compositions, and proved several interesting properties using combinatorial arguments. Now, jointly with Daniel Checa, we have obtained several beautiful and new results using generating functions! (see our paper in INTEGERS).
(03/2024) With Rigo, we introduced both symmetric and asymmetric statistic in Dyck paths. Now, in collaboration with Jean-Luc, we have investigated this interesting parameter in Dyck paths with air pocktes. Our results have been published in the paper titled Symmetries in Dyck paths with air pockets (in Aequationes Mathematice).
(02/2024) We have completed our study on classical and consecutive pattern avoidance in Catalan words:
In our paper titled Last symbol distribution in pattern avoiding Catalan words with Jean-Luc Baril and Javier Gonzalez, published in Mathematics for Computer Sciences, we have enumerated the last symbol statistic on the sets of Catalan words avoiding a pattern of lenght at most three. We end this paper by describing a computational approach using computer algebra.
In our second paper, titled Consecutive pattern-avoidance in Catalan words according to the last symbol co-authored with Jean-Luc Baril, Daniela Colmenanres, Emmanuel Silva, Lina Simbaqueba and Diana Toquica, and published in RAIRO-Theoretical Informatics and Applications, we investigate the distribution of the last symbol statistic on the sets of Catalan words avoiding a consecutive pattern of length at most three.
We have also achieved a comprehensive classification of avoiding ordered pair relations and descent distribution in Catalan words. See the paper Descent distribution on Catalan words avoiding ordered pairs of relations, with Jean-Luc Baril, and published in Advances in Applied Mathematics.
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