The Partial Partition Complex

Abstract

The set of partial partitions of $\{1,\ldots,n\}$, ordered by containment, forms an abstract simplicial complex $D_n$ whose vertices are the nonempty subsets of $\{1,\ldots,n\}$ and whose simplices are collections of pairwise disjoint subsets. We prove that $D_n$ is vertex-decomposable, give an explicit nonpure shelling, and use it to compute the reduced homology: for $1 \le j \le n$, the homology in dimension $j-1$ is free abelian of rank equal to the number of partitions of $\{1,\ldots,n\}$ into $j$ blocks containing no singleton blocks. Explicit generators are constructed as boundary complexes of $j$-dimensional cross-polytopes, one for each non-singleton partition. We also prove that the automorphism group of $D_n$ is the symmetric group on $n$ letters.