Distribution of genus among numerical semigroups with fixed Frobenius number

TL;DR

This paper analyzes the distribution of genus among numerical semigroups with a fixed Frobenius number, revealing that most have genus near three-fourths of the Frobenius number and providing asymptotic distribution insights.

Contribution

It derives the asymptotic distribution of genus in numerical semigroups with fixed Frobenius number and establishes monotonicity properties of their counts.

Findings

Genus distribution approximates a Gaussian times a power series.

Most semigroups have genus close to 3/4 of the Frobenius number.

Number of semigroups with fixed Frobenius number increases with f, specifically N(f)<N(f+2).

Abstract

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. The size of its complement is called the genus and the largest number in the complement is called its Frobenius number. We consider the set of numerical semigroups with a fixed Frobenius number $f$ and analyse their genus. We find the asymptotic distribution of genus in this set of numerical semigroups and show that it is a product of a Gaussian and a power series. We show that almost all numerical semigroups with Frobenius number $f$ have genus close to $\frac{3f}{4}$. We denote the number of numerical semigroups of Frobenius number $f$ by $N(f)$. While $N(f)$ is not monotonic we prove that $N(f)<N(f+2)$ for every $f$.