Numerical semigroups with concentration two

TL;DR

This paper studies numerical semigroups with a specific concentration value of two, providing algorithms for their classification based on key parameters and proving they satisfy Wilf's conjecture.

Contribution

It introduces the class of numerical semigroups with concentration two, offers algorithms for their enumeration, and proves they satisfy Wilf's conjecture.

Findings

Algorithms to compute all semigroups with given parameters

Semigroups with concentration two satisfy Wilf's conjecture

Characterization of semigroups with concentration two

Abstract

We define the concentration of a numerical semigroup $S$ as $\mathsf{C}(S)=\max \left\{\text{next}_S(s)-s ~|~ s\in S \backslash \{0\}\right\}$ wherein $\text{next}_S(s)=\min\left\{x \in S ~|~ s<x\right\}$. In this paper, we study the class of numerical semigroups with concentration $2$. We give algorithms to calculate the whole set of this class of semigroups with given multiplicity, genus or Frobenius number. Separately, we prove that this class of semigroups verifies the Wilf's conjecture.