{
  "id": "1008.2950",
  "version": "v1",
  "published": "2010-08-17T19:23:18.000Z",
  "updated": "2010-08-17T19:23:18.000Z",
  "title": "Partitions, rooks, and symmetric functions in noncommuting variables",
  "authors": [
    "Mahir Bilen Can",
    "Bruce E. Sagan"
  ],
  "comment": "8 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\Pi_n$ denote the set of all set partitions of $\\{1,2,\\ldots,n\\}$. We consider two subsets of $\\Pi_n$, one connected to rook theory and one associated with symmetric functions in noncommuting variables. Let $\\cE_n\\sbe\\Pi_n$ be the subset of all partitions corresponding to an extendable rook (placement) on the upper-triangular board, $\\cT_{n-1}$. Given $\\pi\\in\\Pi_m$ and $\\si\\in\\Pi_n$, define their {\\it slash product\\/} to be $\\pi|\\si=\\pi\\cup(\\si+m)\\in\\Pi_{m+n}$ where $\\si+m$ is the partition obtained by adding $m$ to every element of every block of $\\si$. Call $\\tau$ {\\it atomic\\/} if it can not be written as a nontrivial slash product and let $\\cA_n\\sbe\\Pi_n$ denote the subset of atomic partitions. Atomic partitions were first defined by Bergeron, Hohlweg, Rosas, and Zabrocki during their study of $NCSym$, the symmetric functions in noncommuting variables. We show that, despite their very different definitions, $\\cE_n=\\cA_n$ for all $n\\ge0$. Furthermore, we put an algebra structure on the formal vector space generated by all rook placements on upper triangular boards which makes it isomorphic to $NCSym$. We end with some remarks and an open problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-08-17T19:23:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A18",
      "05E05"
    ],
    "keywords": [
      "symmetric functions",
      "noncommuting variables",
      "atomic partitions",
      "nontrivial slash product",
      "formal vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.2950B"
    }
  }
}