{
  "id": "2312.02830",
  "version": "v1",
  "published": "2023-12-05T15:25:57.000Z",
  "updated": "2023-12-05T15:25:57.000Z",
  "title": "Differential operators, grammars and Young tableaux",
  "authors": [
    "Shi-Mei Ma",
    "Jean Yeh",
    "Yeong-Nan Yeh"
  ],
  "comment": "38 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In algebraic combinatorics and formal calculation, context-free grammar is defined by a formal derivative based on a set of substitution rules. In this paper, we investigate this issue from three related viewpoints. Firstly, we introduce a differential operator method. As one of the applications, we deduce a new grammar for the Narayana polynomials. Secondly, we investigate the normal ordered grammars associated with the Eulerian polynomials. Thirdly, motivated by the theory of differential posets, we introduce a box sorting algorithm which leads to a bijection between the terms in the expansion of $(cD)^nc$ and a kind of ordered weak set partitions, where $c$ is a smooth function in the indeterminate $x$ and $D$ is the derivative with respect to $x$. Using a map from ordered weak set partitions to standard Young tableaux, we find an expansion of $(cD)^nc$ in terms of standard Young tableaux. Combining this with the theory of context-free grammars, we provide a unified interpretations for the Ramanujan polynomials, Andr\\'e polynomials, left peak polynomials, interior peak polynomials, Eulerian polynomials of types $A$ and $B$, $1/2$-Eulerian polynomials, second-order Eulerian polynomials, and Narayana polynomials of types $A$ and $B$ in terms of standard Young tableaux. Along the same lines, we present an expansion of the powers of $c^kD$ in terms of standard Young tableaux, where $k$ is a positive integer. In particular, we provide four interpretations for the second-order Eulerian polynomials. All of the above apply to the theory of formal differential operator rings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-05T15:25:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19",
      "05E10"
    ],
    "keywords": [
      "standard young tableaux",
      "ordered weak set partitions",
      "second-order eulerian polynomials",
      "narayana polynomials",
      "formal differential operator rings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}