Ordered set partition posets
Bruce E Sagan (Michigan State University), Sheila Sundaram (University of Minnesota)

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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Combinatorial Mathematics · Finite Group Theory Research · Advanced Mathematical Identities
