On the enumeration of the set of numerical semigroups with fixed Frobenius number and fixed number of second kind gaps

TL;DR

This paper investigates the relationship between invariants of numerical semigroups and their second kind gaps, and introduces an algorithm to enumerate all semigroups with a given Frobenius number and specific second kind gaps.

Contribution

It provides a novel algorithm for enumerating numerical semigroups with fixed Frobenius number and second kind gaps, linking invariants to gap counts.

Findings

Established relationships between invariants and second kind gaps

Developed an algorithm for enumeration based on fixed parameters

Enabled systematic enumeration of semigroups with specified properties

Abstract

We study how certain invariants of numerical semigroups relate to the number of second kind gaps. Furthermore, given two fixed non-negative integers F and k, we provide an algorithm to compute all the numerical semigroups whose Frobenius number is F and which have exactly k second kind gaps.