# The tree of numerical semigroups with low multiplicity

## TL;DR

This paper investigates the growth of the number of numerical semigroups with fixed low multiplicity and genus, utilizing the Kunz polytope and partition problems to establish increasing trends.

## Contribution

It demonstrates that the count of numerical semigroups with multiplicity three, four, or five increases with genus, and provides explicit enumeration for multiplicity four using partition conditions.

## Key findings

- Number of semigroups increases with genus for multiplicities 3, 4, 5.
- Explicit partition-based formula for multiplicity four semigroups.
- Use of Kunz polytope to relate semigroup counting to integer partitions.

## Abstract

We show that the number of numerical semigroups with multiplicity three, four or five and fixed genus is increasing as a function in the genus. To this end we use the Kunz polytope for these multiplicities. Counting numerical semigroups with fixed multiplicity and genus is then an integer partition problem with some extra conditions (those of membership to the Kunz polytope). For the particular case of multiplicity four, we are able to prove that the number of numerical semigroups with multiplicity four and genus $g$ is the number of partitions $x+y+z=g+6$ with $0<x\le y\le z$, $x\neq 1$, $y\neq 2$ and $z\neq 3$.