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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.11190 |
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| _version_ | 1866914620960145408 |
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| author | Gottstein, Michael J. |
| author_facet | Gottstein, Michael J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_11190 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Partial Partition Complex Gottstein, Michael J. Combinatorics 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. |
| title | The Partial Partition Complex |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2311.11190 |