{"paper":{"title":"Filters in the partition lattice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dustin Hedmark, Richard Ehrenborg","submitted_at":"2016-06-05T01:58:11Z","abstract_excerpt":"Given a filter $\\Delta$ in the poset of compositions of $n$, we form the filter $\\Pi^{*}_{\\Delta}$ in the partition lattice. We determine all the reduced homology groups of the order complex of $\\Pi^{*}_{\\Delta}$ as ${\\mathfrak S}_{n-1}$-modules in terms of the reduced homology groups of the simplicial complex $\\Delta$ and in terms of Specht modules of border shapes. We also obtain the homotopy type of this order complex. These results generalize work of Calderbank--Hanlon--Robinson and Wachs on the $d$-divisible partition lattice. Our main theorem applies to a plethora of examples, including "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.01443","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}