Numerical Semigroups with $a_e = 2g+1$

TL;DR

This paper studies numerical semigroups with the largest possible generator, $2g+1$, revealing their symmetry properties, relation to symmetric semigroups, and positive answer to Wilf's conjecture for these cases.

Contribution

It characterizes semigroups with generator $2g+1$, showing their symmetry and confirming Wilf's conjecture for these semigroups.

Findings

Semigroups with generator $2g+1$ are closely related to symmetric semigroups.

These semigroups exhibit interesting symmetry properties.

Wilf's conjecture holds for these semigroups and some derived semigroups.

Abstract

This article discusses numerical semigroups having a generator which is as large as possible. This turns out to be $2g+1$, where $g$ is the genus of the semigroup. We will show that these semigroups are closely related to symmetric semigroups and have interesting symmetry properties themselves. Furthermore we will show that Wilf's question has a positive answer for these semigroups and some semigroups derived thereof.