Gapsets and the $k$-generalized Fibonacci sequences

TL;DR

This paper introduces a new combinatorial approach to studying gapsets and $m$-extensions, connecting them to tilings and proving conjectures related to their enumeration, with explicit formulas for certain cases.

Contribution

It generalizes the concept of Kunz coordinates to gapsets and $m$-extensions, proving a version of Bras-Amorós conjecture and providing bounds and explicit formulas for counting gapsets.

Findings

Proved a version of Bras-Amorós conjecture for $m$-extensions.

Established bounds on the number of gapsets with fixed genus and depth.

Derived explicit formulas for gapsets with multiplicity 3 or 4.

Abstract

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-Amor\'{o}s 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.