(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).

(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).

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