Computation of numerical semigroups by means of seeds

TL;DR

This paper introduces the concept of seeds in numerical semigroups to efficiently explore their structure and develop a new algorithm for computing and counting semigroups of a given genus.

Contribution

It presents a novel seed-based approach that simplifies the exploration of the semigroup tree and improves the computation of numerical semigroups.

Findings

Seeds enable efficient traversal of the semigroup tree

New algorithm effectively computes semigroups of a specified genus

Approach broadens understanding of numerical semigroup structure

Abstract

For the elements of a numerical semigroup which are larger than the Frobenius number, we introduce the definition of, seed, by broadening the notion of generator. This new concept allows us to explore the semigroup tree in an alternative efficient way, since the seeds of each descendant can be easily obtained from the seeds of its parent. The paper is devoted to presenting the results which are related to this approach, leading to a new algorithm for computing and counting the semigroups of a given genus.