Sub-Fibonacci behavior in numerical semigroup enumeration

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 $S{\phi }^g$, where $S$ is some positive constant and $\phi \approx 1.61803$ 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-Amorós (2008) that $n_g\ge n_{g-1}+n_{g-2}$. Our main tools are the use of Kunz coordinates, introduced by Kunz (1987), and a result by Zhao (2011) bounding weighted graph homomorphisms.

20 pages, 2 figures, 2 tables; version published in Comb. Theory; code available at https://github.com/zhdag/stressed

References in corpus (1)

• Degree asymptotics of the numerical semigroup tree

Cited by in corpus (3)

• Ordinarization numbers of numerical semigroups
• Kunz languages for numerical semigroups are context sensitive
• Counting numerical semigroups by Frobenius number, multiplicity, and depth