Enumerating lattices of subsets

Abstract

Given k sets such that no one is contained in another, there is an associated lattice on the power set P([k]) corresponding to inclusion relations among unions of the sets. Two lattices on P([k]) are equivalent if there is a permutation of [k] under which they correspond. We show that for k=1, 2, 3, and 4, there are 1, 1, 4, and 50 equivalence classes of lattices on P([k]) obtained from sets in this way. We cannot find a reference to previous work on this enumeration problem in the literature, and so wish to introduce it for subsequent investigation. We explain how the problem arose from algebraic topology.