The set of numerical semigroups of a given genus

TL;DR

This paper introduces a novel method for constructing numerical semigroups of a fixed genus by leveraging their Frobenius number and a tree structure, enhanced by an algorithm using Kunz-coordinates vectors.

Contribution

It presents a new approach to generate numerical semigroups of a given genus using Frobenius number and a tree-based structure, improving efficiency with Kunz-coordinates.

Findings

Defined an equivalence relation over semigroups with fixed Frobenius number and genus.

Established a tree structure for each equivalence class.

Developed a more efficient algorithm using Kunz-coordinates vectors.

Abstract

In this paper we present a new approach to construct the set of numerical semigroups with a fixed genus. Our methodology is based on the construction of the set of numerical semigroups with fixed Frobenius number and genus. An equivalence relation is given over this set and a tree structure is defined for each equivalence class. We also provide a more efficient algorithm based in the translation of a numerical semigroup to its so-called Kunz-coordinates vector.