Degree asymptotics of the numerical semigroup tree

Abstract

A \emph{numerical semigroup} is a subset $\Lambda$ of the nonnegative integers that is closed under addition, contains $0$, and omits only finitely many nonnegative integers (called the \emph{gaps} of $\Lambda$). The collection of all numerical semigroups may be visually represented by a tree of element removals, in which the children of a semigroup $\Lambda$ are formed by removing one element of $\Lambda$ that exceeds all existing gaps of $\Lambda$. 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 $\phi^{-h-2}$ of all numerical semigroups, where $\phi$ is the golden ratio.