{
  "id": "math/0504164",
  "version": "v7",
  "published": "2005-04-08T11:36:14.000Z",
  "updated": "2006-11-27T15:25:09.000Z",
  "title": "Log-concavity and LC-positivity",
  "authors": [
    "Yi Wang",
    "Yeong-Nan Yeh"
  ],
  "comment": "16 pages",
  "journal": "J. Combin. Theory, Ser. A 114 (2007) 195--210",
  "doi": "10.1016/j.jcta.2006.02.001",
  "categories": [
    "math.CO"
  ],
  "abstract": "A triangle $\\{a(n,k)\\}_{0\\le k\\le n}$ of nonnegative numbers is LC-positive if for each $r$, the sequence of polynomials $\\sum_{k=r}^{n}a(n,k)q^k$ is $q$-log-concave. It is double LC-positive if both triangles $\\{a(n,k)\\}$ and $\\{a(n,n-k)\\}$ are LC-positive. We show that if $\\{a(n,k)\\}$ is LC-positive then the log-concavity of the sequence $\\{x_k\\}$ implies that of the sequence $\\{z_n\\}$ defined by $z_n=\\sum_{k=0}^{n}a(n,k)x_k$, and if $\\{a(n,k)\\}$ is double LC-positive then the log-concavity of sequences $\\{x_k\\}$ and $\\{y_k\\}$ implies that of the sequence $\\{z_n\\}$ defined by $z_n=\\sum_{k=0}^{n}a(n,k)x_ky_{n-k}$. Examples of double LC-positive triangles include the constant triangle and the Pascal triangle. We also give a generalization of a result of Liggett that is used to prove a conjecture of Pemantle on characteristics of negative dependence.",
  "revisions": [
    {
      "version": "v7",
      "updated": "2006-11-27T15:25:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A20",
      "15A04"
    ],
    "keywords": [
      "log-concavity",
      "lc-positivity",
      "double lc-positive",
      "constant triangle",
      "pascal triangle"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......4164W"
    }
  }
}