Homology of Partial Partitions Ordered by Inclusion

TL;DR

This paper explores the topological structure of partial partitions of finite sets ordered by inclusion, revealing their homological properties and basis through combinatorial and algebraic topology methods.

Contribution

It introduces a novel analysis of the homology of the complex of partial partitions using shellable nonpure complexes and combinatorial cross-polytopes.

Findings

Betti numbers of the complex are explicitly computed.

Homology basis consists of boundaries of combinatorial cross-polytopes.

The structure is shown to be shellable and nonpure.

Abstract

We investigate the set of partial partitions of a finite set, ordered by inclusion. With this ordering the set of partial partitions can be studied as an abstract simplicial complex. We use the theory of shellable nonpure complexes to find its Betti numbers and prove that there is a basis of its homology that consists of boundaries of combinatorial cross-polytopes of various dimensions.