{
  "id": "1701.04321",
  "version": "v1",
  "published": "2017-01-16T15:16:15.000Z",
  "updated": "2017-01-16T15:16:15.000Z",
  "title": "Proof of an entropy conjecture of Leighton and Moitra",
  "authors": [
    "Hüseyin Acan",
    "Pat Devlin",
    "Jeff Kahn"
  ],
  "comment": "9 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove the following conjecture of Leighton and Moitra. Let $T$ be a tournament on $[n]$ and $S_n$ the set of permutations of $[n]$. For an arc $uv$ of $T$, let $A_{uv}=\\{\\sigma \\in S_n \\, : \\, \\sigma(u)<\\sigma(v) \\}$. $\\textbf{Theorem.}$ For a fixed $\\varepsilon>0$, if $\\mathbb{P}$ is a probability distribution on $S_n$ such that $\\mathbb{P}(A_{uv})>1/2+\\varepsilon$ for every arc $uv$ of $T$, then the binary entropy of $\\mathbb{P}$ is at most $(1-\\vartheta_{\\varepsilon})\\log_2 n!$ for some (fixed) positive $\\vartheta_\\varepsilon$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-16T15:16:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C20",
      "05D40",
      "94A17",
      "06A07"
    ],
    "keywords": [
      "entropy conjecture",
      "probability distribution",
      "binary entropy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}