Numerical Semigroups of small and large type

TL;DR

This paper investigates the properties and classifications of numerical semigroups, focusing on their type, Frobenius number, and genus, and introduces new characterizations and asymptotic counting results.

Contribution

It provides explicit characterizations of numerical semigroups with specific type values and shows that the counts of certain semigroup classes become constant for large parameters.

Findings

Characterization of semigroups with t=2g-F as almost symmetric

Explicit description of semigroups with t=g/(F+1-g)

Counts of semigroups with fixed parameters stabilize asymptotically

Abstract

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number F, genus g and type t. It is known that for any numerical semigroup $\frac{g}{F+1-g}\leq t\leq 2g-F$. Numerical semigroups with $t=2g-F$ are called almost symmetric, we introduce a new property that characterises them. We give an explicit characterisation of numerical semigroups with $t=\frac{g}{F+1-g}$. We show that for a fixed $\alpha$ the number of numerical semigroups with Frobenius number $F$ and type $F-\alpha$ is eventually constant for large $F$. Also the number of numerical semigroups with genus $g$ and type $g-\alpha$ is also eventually constant for large $g$.