Ordered set partition posets

The lattice of partitions of a set and its d-divisible generalization have been much studied for their combinatorial, topological, and representation-theoretic properties. An ordered set partition is a set partition where the subsets are listed in a specific order. Ordered set partitions appear in combinatorics, number theory, permutation polytopes, and the study of coinvariant algebras. The ordered set partitions of {1,\ldots,n} can be partially ordered by refinement and then a unique minimal element attached, resulting in a lattice Omega_n. This lattice has appeared while studying other combinatorial objects, but not as the central focus. The purpose of this paper is to provide the first comprehensive look at Omega_n. In particular, we show that it admits a recursive atom ordering, and study the action of the symmetric group S_n on associated homology groups, looking in particular at the multiplicity of the trivial representation. We also consider the related posets where every block has size either divisible by some fixed d at least 2 or congruent to 1 modulo d. Open problems and avenues for future research are scattered throughout.