# The set of numerical semigroups of a given multiplicity and Frobenius number

## TL;DR

This paper investigates the structure of numerical semigroups with fixed multiplicity and Frobenius number, providing an algorithm to enumerate all such semigroups and applying it to compute semigroups with a given multiplicity and genus.

## Contribution

It introduces an algorithmic approach to classify numerical semigroups with specified parameters, advancing understanding of their structural properties.

## Key findings

- Developed an algorithm to enumerate all numerical semigroups with fixed multiplicity and Frobenius number.
- Successfully computed the set of all numerical semigroups with given multiplicity and genus.
- Enhanced the understanding of the structure of numerical semigroups with specific invariants.

## Abstract

We study the structure of the family of numerical semigroups with fixed multiplicity and Frobenius number. We give an algorithmic method to compute all the semigroups in this family. As an application we compute the set of all numerical semigroups with given multiplicity and genus.