{
  "id": "1309.5527",
  "version": "v3",
  "published": "2013-09-21T19:56:38.000Z",
  "updated": "2015-04-04T18:23:50.000Z",
  "title": "On the (co)homology of the poset of weighted partitions",
  "authors": [
    "Rafael S. González D'León",
    "Michelle L. Wachs"
  ],
  "comment": "50 pages, final version, to appear in Trans. AMS",
  "categories": [
    "math.CO"
  ],
  "abstract": "We consider the poset of weighted partitions $\\Pi_n^w$, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of $\\Pi_n^w$ provide a generalization of the lattice $\\Pi_n$ of partitions, which we show possesses many of the well-known properties of $\\Pi_n$. In particular, we prove these intervals are EL-shellable, we show that the M\\\"obius invariant of each maximal interval is given up to sign by the number of rooted trees on on node set $\\{1,2,\\dots,n\\}$ having a fixed number of descents, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted $\\mathfrak{S}_n$-module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of $\\Pi_n^w$ has a nice factorization analogous to that of $\\Pi_n$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-12-14T02:58:50.000Z",
      "comment": "48 pages, minor revisions were made to the introduction of the previous version",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-04-04T18:23:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45",
      "05E18",
      "05A18",
      "17B01"
    ],
    "keywords": [
      "weighted partitions",
      "maximal interval",
      "free lie algebra",
      "dual operads",
      "well-known properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.5527G"
    }
  }
}