{
  "id": "1904.06630",
  "version": "v1",
  "published": "2019-04-14T05:19:22.000Z",
  "updated": "2019-04-14T05:19:22.000Z",
  "title": "Flagged $(\\mathcal{P},ρ)$-partitions",
  "authors": [
    "Sami Assaf",
    "Nantel Bergeron"
  ],
  "comment": "18 pages, 10 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We introduce the theory of $(\\mathcal{P},\\rho)$-partitions, depending on a poset $\\mathcal{P}$ and a map $\\rho$ from $\\mathcal{P}$ to positive integers. The generating function $\\mathfrak{F}_{\\mathcal{P},\\rho}$ of $(\\mathcal{P},\\rho)$-partitions is a polynomial that, when the images of $\\rho$ tend to infinity, tends to Stanley's generating function $F_{\\mathcal{P}}$ of $\\mathcal{P}$-partitions. Analogous to Stanley's fundamental theorem for $\\mathcal{P}$-partitions, we show that the set of $(\\mathcal{P},\\rho)$-partitions decomposes as a disjoint union of $(\\mathcal{L},\\rho)$-partitions where $\\mathcal{L}$ runs over the set of linear extensions of $\\mathcal{P}$. In this more general context, the set of all $\\mathfrak{F}_{\\mathcal{L},\\rho}$ for linear orders $\\mathcal{L}$ over determines a basis of polynomials. We thus introduce the notion of flagged $(\\mathcal{P},\\rho)$-partitions, and we prove that the set of all $\\mathfrak{F}_{\\mathcal{L},\\rho}$ for flagged $(\\mathcal{L},\\rho)$-partitions for linear orders $\\mathcal{L}$ is precisely the fundamental slide basis of the polynomial ring, introduced by the first author and Searles. Our main theorem shows that any generating function $\\mathfrak{F}_{\\mathcal{P},\\rho}$ of flagged $(\\mathcal{P},\\rho)$-partitions is a positive integer linear combination of slide polynomials. As applications, we give a new proof of positivity of the slide product and, motivating our nomenclature, we also prove flagged Schur functions are slide positive.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-14T05:19:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomial",
      "linear orders",
      "positive integer linear combination",
      "stanleys fundamental theorem",
      "fundamental slide basis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}