  A numerical set $T$ is a subset of $\mathbb N_0$ that contains $0$ and has
finite complement. The atom monoid of $T$ is the set of $x \in \mathbb N_0$
such that $x+T \subseteq T$. Marzuola and Miller introduced the anti-atom
problem: how many numerical sets have a given atom monoid? This is equivalent
to asking for the number of integer partitions with a given set of hook
lengths. We introduce the void poset of a numerical semigroup $S$ and show that
numerical sets with atom monoid $S$ are in bijection with certain order ideals
of this poset. We use this characterization to answer the anti-atom problem
when $S$ has small type.
