{
  "id": "2502.02262",
  "version": "v2",
  "published": "2025-02-04T12:20:03.000Z",
  "updated": "2025-06-24T17:08:32.000Z",
  "title": "The major index (maj) and its Schützenberger dual",
  "authors": [
    "Oleg Ogievetsky",
    "Senya Shlosman"
  ],
  "comment": "20 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We construct the independent particle representation for the Semistandard Young Tableaux (SsYT) of skew shape $\\lambda/\\mu.$ The partition function of this particle system gives the generating function of the SsYT of skew shape $\\lambda/\\mu.$ Thus we obtain a bijective proof of the Stanley formula for the SsYT generating function. To do this we define for every SsYT $T$ its plinth, $\\mathsf{p}\\left( T\\right) ,$ which is a SsYT of the same shape $\\lambda/\\mu.$ The set of plinths is finite. Our bijection associates to every SsYT $T$ a pair $\\left( \\mathsf{p}\\left( T\\right) ,Y\\left( T-\\mathsf{p}\\left( T\\right) \\right) \\right) ,$ where $Y\\left( T-\\mathsf{p}\\left( T\\right) \\right) $ is the reading Young diagram of the SsYT $\\left( T-\\mathsf{p}\\left( T\\right) \\right) $. \\newline In particular, every Standard Young Tableau (SYT) $P$ has its plinth, $\\mathsf{p}\\left( P\\right) $. The two statistics of SYT-s -- the volume $\\left\\vert \\mathsf{p}\\left( P\\right) \\right\\vert $ and $\\mathsf{maj}\\left( P\\right) $ -- are related via the Sch\\\"{u}tzenberger involution $Sch:$% \\[ \\left\\vert \\mathsf{p}\\left( P\\right) \\right\\vert =\\mathsf{maj}\\left( Sch\\left( P\\right) \\right) . \\]",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-06-24T17:08:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19",
      "05A15"
    ],
    "keywords": [
      "schützenberger dual",
      "major index",
      "skew shape",
      "generating function",
      "independent particle representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}