Counting Ideals in Numerical Semigroups

Abstract

If $S$ is a numerical semigroup, let $m(S,k)$ denote the number of ideals of $S$ with codimension $k$ and let $n(S,k)$ denote the number of ideals of $S$ with conductor $k$. We compute the generating function of the sequence $m(S,k)$ for all numerical semigroups of embedding dimension $2$ and for $S=⟨3,n+2,2n+1⟩$. We also prove that the sequence $n(S,k)$ becomes stationary after a certain term and compute the stationary terms for numerical semigroups of the form $⟨a,a+1⟩$.