{
  "id": "1606.01443",
  "version": "v1",
  "published": "2016-06-05T01:58:11.000Z",
  "updated": "2016-06-05T01:58:11.000Z",
  "title": "Filters in the partition lattice",
  "authors": [
    "Richard Ehrenborg",
    "Dustin Hedmark"
  ],
  "comment": "29 pages, 1 figures and 2 tables",
  "categories": [
    "math.CO"
  ],
  "abstract": "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 filters associated to integer knapsack partitions and filters generated by all partitions having block sizes $a$ or~$b$. We also obtain the reduced homology groups of the filter generated by all partitions having block sizes belonging to the arithmetic progression $a, a + d, \\ldots, a + (a-1) \\cdot d$, extending work of Browdy.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-05T01:58:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E25",
      "05A18",
      "06A11"
    ],
    "keywords": [
      "partition lattice",
      "reduced homology groups",
      "block sizes",
      "order complex",
      "integer knapsack partitions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}