Degree asymptotics of the numerical semigroup tree

Abstract

A \emph{numerical semigroup} is a subset $Λ$ of the nonnegative integers that is closed under addition, contains $0$, and omits only finitely many nonnegative integers (called the \emph{gaps} of $Λ$). The collection of all numerical semigroups may be visually represented by a tree of element removals, in which the children of a semigroup $Λ$ are formed by removing one element of $Λ$ that exceeds all existing gaps of $Λ$. In general, a semigroup may have many children or none at all, making it difficult to understand the number of semigroups at a given depth on the tree. We investigate the problem of estimating the number of semigroups at depth $g$ (i.e.\ of genus $g$) with $h$ children, showing that as $g$ becomes large, it tends to a proportion $φ^{-h-2}$ of all numerical semigroups, where $φ$ is the golden ratio.

Figures (1)

• Figure 1: Tree showing the numerical semigroups of genus $g \leq 4$

Theorems & Definitions (27)

• Theorem 1: 7; conjectured by Bras-Amorós in 1
• Conjecture 1: 4
• Conjecture 2: Conjecture 2 in 6
• Theorem 2
• Lemma 1: Lemma 7 from 7
• Lemma 2: Lemma 8 from 7
• Lemma 3: from Sect. 3.2, page 10 of 7
• proof
• Lemma 4
• proof