On the enumeration of the set of atomic numerical semigroups with fixed Frobenius number

TL;DR

This paper introduces an algorithm to enumerate all atomic numerical semigroups with a given Frobenius number, generalizing previous methods for irreducible semigroups to a broader class.

Contribution

The paper presents a novel algorithm for enumerating all atomic numerical semigroups with a fixed Frobenius number, extending prior work on irreducible semigroups.

Findings

Algorithm successfully enumerates all atomic numerical semigroups for given Frobenius numbers.

Extends the classification from irreducible to atomic numerical semigroups.

Provides a computational tool for studying the structure of numerical semigroups.

Abstract

A numerical semigroup is irreducible if it cannot be obtained as intersection of two numerical semigroups containing it properly. If we only consider numerical semigroups with the same Frobenius number, that concept is generalized to atomic numerical semigroup. Based on a previous one developed to obtain all irreducible numerical semigroups with a fixed Frobenius number, we present an algorithm to compute all atomic numerical semigroups with a fixed Frobenius number.