2022/08/16 by Gilberto B. Almeida Filho, Filho, Gilberto B. Almeida, Matheus Bernardini +1
Computer Science · Mathematics · #05A19 (Secondary) #20M14 (Primary) 05A15 #Advanced Combinatorial Mathematics #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2208.07692
openalex publication_date 2022/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we bring the terminology of the Kunz coordinates of numerical semigroups to gapsets and we generalize this concept to m-extensions. It allows us to identify gapsets and, in general, m-extensions with tilings of boards. As a consequence, we prove a version of Bras-Amorós conjecture for m-extensions. Besides, we obtain a lower bound for the number of gapsets with fixed genus and depth at most 3 and a family of upper bounds for the number of gapsets with fixed genus. Moreover, we present explicit formulas for the number of gapsets with fixed genus and depth, when the multiplicity is 3 or 4, and, in some cases, for the number of gapsets with fixed genus and depth.