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We investigate the set of partial partitions of a finite set, ordered byinclusion. With this ordering the set of partial partitions can be studied asan abstract simplicial complex. We use the theory of shellable nonpurecomplexes to find its Betti numbers and prove that there is a basis of itshomology that consists of boundaries of combinatorial cross-polytopes ofvarious dimensions.

Homology of Partial Partitions Ordered by Inclusion