Fibonacci-like growth of numerical semigroups of a given genus

TL;DR

This paper establishes that the number of numerical semigroups of a given genus grows asymptotically like a constant times the golden ratio to the power of the genus, and explores properties related to their structure.

Contribution

It provides the first asymptotic estimate for the count of numerical semigroups of a fixed genus, confirming conjectures about their exponential growth rate.

Findings

Number of semigroups grows like S * φ^g, with φ as the golden ratio.

Proportion of semigroups with f < 3m approaches 1 as genus g increases.

Resolved conjectures related to the growth of numerical semigroups.

Abstract

We give an asymptotic estimate of the number of numerical semigroups of a given genus. In particular, if $n_g$ is the number of numerical semigroups of genus $g$, we prove that $n_g$ tends to $S \phi^g$, where $\phi$ is the golden ratio, and $S$ is a constant, resolving several related conjectures concerning the growth of $n_g$. In addition, we show that the proportion of numerical semigroups of genus $g$ satisfying $f < 3m$ approaches 1 as $g \rightarrow \infty$, where $m$ is the multiplicity and $f$ is the Frobenius number.