Sub-Fibonacci behavior in numerical semigroup enumeration

TL;DR

This paper investigates the enumeration of numerical semigroups, providing bounds on deviations from exponential growth and exploring implications for a conjecture related to their counting sequence.

Contribution

It introduces exponential bounds on factors affecting the growth of numerical semigroups and applies Kunz coordinates and graph homomorphism bounds to analyze their enumeration.

Findings

Established sharp asymptotic bounds on the number of numerical semigroups of a given genus.

Identified the role of depth in the enumeration and growth deviations.

Provided insights supporting Bras-Amorós' conjecture on the sequence of counts.

Abstract

In 2013, Zhai proved that most numerical semigroups of a given genus have depth at most $3$ and that the number $n_g$ of numerical semigroups of a genus $g$ is asymptotic to $S\varphi^g$, where $S$ is some positive constant and $\varphi \approx 1.61803$ is the golden ratio. In this paper, we prove exponential upper and lower bounds on the factors that cause $n_g$ to deviate from a perfect exponential, including the number of semigroups with depth at least $4$. Among other applications, these results imply the sharpest known asymptotic bounds on $n_g$ and shed light on a conjecture by Bras-Amor\'os (2008) that $n_g \geq n_{g-1} + n_{g-2}$. Our main tools are the use of Kunz coordinates, introduced by Kunz (1987), and a result by Zhao (2011) bounding weighted graph homomorphisms.