Ordered set partition posets

TL;DR

This paper introduces and thoroughly analyzes the lattice of ordered set partitions, exploring its recursive structure, group actions, and connections to other combinatorial objects, providing new insights into its algebraic and topological properties.

Contribution

It provides the first comprehensive study of the lattice Omega_n of ordered set partitions, including recursive atom orderings and symmetric group actions on homology.

Findings

Omega_n admits a recursive atom ordering

Symmetric group S_n acts on homology groups with identifiable multiplicities

Connections to other combinatorial objects are established

Abstract

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.