{
  "id": "2308.10100",
  "version": "v1",
  "published": "2023-08-19T20:13:08.000Z",
  "updated": "2023-08-19T20:13:08.000Z",
  "title": "Catalan numbers: from FC elements to classical diagram algebras",
  "authors": [
    "Sadek Al Harbat"
  ],
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "Let $W^c(A_n)$ be the set of fully commutative elements in the $A_n$-type Coxeter group. Using only the settings of their canonical form, we recount $W^c(A_n)$ by the definition of the Catalan numbers $C_{n+1}$ and find the Narayana numbers as well as the Catalan triangle. We determine the unique bijection between $W^c(A_n)$ and the set of non crossing diagrams that respects the monomial diagrammatical multiplication in the $A_n$-type Temperley-Lieb algebra.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-19T20:13:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "05Axx"
    ],
    "keywords": [
      "classical diagram algebras",
      "catalan numbers",
      "fc elements",
      "type temperley-lieb algebra",
      "type coxeter group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}