{
  "id": "1911.12791",
  "version": "v1",
  "published": "2019-11-28T17:03:47.000Z",
  "updated": "2019-11-28T17:03:47.000Z",
  "title": "Partition and Cohen-Macaulay Extenders",
  "authors": [
    "Joseph Doolittle",
    "Bennet Goeckner",
    "Alexander Lazar"
  ],
  "comment": "12 pages, 4 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "If a pure simplicial complex is partitionable, then its $h$-vector has a combinatorial interpretation in terms of any partitioning of the complex. Given a non-partitionable complex $\\Delta$, we construct a complex $\\Gamma \\supseteq \\Delta$ of the same dimension such that both $\\Gamma$ and the relative complex $(\\Gamma,\\Delta)$ are partitionable. This allows us to rewrite the $h$-vector of any pure simplicial complex as the difference of two $h$-vectors of partitionable complexes, giving an analogous interpretation of the $h$-vector of a non-partitionable complex. By contrast, for a given complex $\\Delta$ it is not always possible to find a complex $\\Gamma$ such that both $\\Gamma$ and $(\\Gamma,\\Delta)$ are Cohen-Macaulay. We characterize when this is possible, and we show that the construction of such a $\\Gamma$ in this case is remarkably straightforward. We end with a note on a similar notion for shellability and a connection to Simon's conjecture on extendable shellability for uniform matroids.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-28T17:03:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45",
      "13F55"
    ],
    "keywords": [
      "cohen-macaulay extenders",
      "pure simplicial complex",
      "non-partitionable complex",
      "combinatorial interpretation",
      "similar notion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}