{
  "id": "1402.1199",
  "version": "v3",
  "published": "2014-02-05T22:04:10.000Z",
  "updated": "2014-03-09T15:47:34.000Z",
  "title": "A combinatorial proof of strict unimodality for $q$-binomial coefficients",
  "authors": [
    "Vivek Dhand"
  ],
  "comment": "7 pages, expanded results",
  "categories": [
    "math.CO"
  ],
  "abstract": "Pak and Panova recently proved that the $q$-binomial coefficient ${m+n \\choose m}_q$ is a strictly unimodal polynomial in $q$ for $m,n \\geq 8$, via the representation theory of the symmetric group. We give a direct combinatorial proof of their result by characterizing when a product of chains is strictly unimodal and then applying O'Hara's structure theorem for the partition lattice $L(m,n)$. In fact, we prove a stronger result: if $m, n \\geq 8d$, and $2d \\leq r \\leq mn/2$, then the $r$-th rank of $L(m,n)$ has at least $d$ more elements that the next lower rank.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-03-09T15:47:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "binomial coefficient",
      "strict unimodality",
      "direct combinatorial proof",
      "applying oharas structure theorem",
      "lower rank"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.1199D"
    }
  }
}