{
  "id": "1806.03509",
  "version": "v1",
  "published": "2018-06-09T17:27:10.000Z",
  "updated": "2018-06-09T17:27:10.000Z",
  "title": "Path counting and rank gaps in differential posets",
  "authors": [
    "Christian Gaetz",
    "Praveen Venkataramana"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We study the gaps $\\Delta p_n$ between consecutive rank sizes in $r$-differential posets by introducing a projection operator whose matrix entries can be expressed in terms of the number of certain paths in the Hasse diagram. We strengthen Miller's result that $\\Delta p_n \\geq 1$, which resolved a longstanding conjecture of Stanley, by showing that $\\Delta p_n \\geq 2r$. We also obtain stronger bounds in the case that the poset has many substructures called threads.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-09T17:27:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "differential posets",
      "rank gaps",
      "path counting",
      "strengthen millers result",
      "matrix entries"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}