# Gapsets of small multiplicity

## TL;DR

This paper characterizes all gapsets with small multiplicity (up to 4) and proves that their count grows monotonically with genus, providing new insights into the structure of numerical semigroups.

## Contribution

It offers a complete classification of gapsets with multiplicity up to 4 and a simplified proof of their counting function's monotonicity.

## Key findings

- Complete classification of gapsets with multiplicity ≤ 4
- Proof that the number of such gapsets is nondecreasing with genus
- Simplified proof of the monotonicity property

## Abstract

A gapset is the complement of a numerical semigroup in N. In this paper, we characterize all gapsets of multiplicity m $\le$ 4. As a corollary, we provide a new simpler proof that the number of gapsets of genus g and fixed multiplicity m $\le$ 4 is a nondecreasing function of g.