A numerical set T is a subset of N0 that contains 0 and has finite complement. The atom monoid of T is the set of x∈N0 such that x+T⊆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.

Since MacWilliams proved the original identity relating the Hamming weight enumerator of a linear code to the weight enumerator of its dual code there have been many different generalizations, leading to the development of $m$-tuple support enumerators. We prove a generalization of theorems of Britz and of Ray-Chaudhuri and Siap, which build on earlier work of Klø ve, Shiromoto, Wan, and others. We then give illustrations of these $m$-tuple weight enumerators.