{
  "id": "2007.01892",
  "version": "v1",
  "published": "2020-07-03T18:13:38.000Z",
  "updated": "2020-07-03T18:13:38.000Z",
  "title": "Generalized Path Pairs and Fuss-Catalan Triangles",
  "authors": [
    "Paul Drube"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Path pairs are a modification of parallelogram polyominoes that provide yet another combinatorial interpretation of the Catalan numbers. More generally, the number of path pairs of length $n$ and distance $\\delta$ corresponds to the $(n-1,\\delta-1)$ entry of Shapiro's so-called Catalan triangle. In this paper, we widen the notion of path pairs $(\\gamma_1,\\gamma_2)$ to the situation where $\\gamma_1$ and $\\gamma_2$ may have different lengths, and then enforce divisibility conditions on runs of vertical steps in $\\gamma_2$. This creates a two-parameter family of integer triangles that generalize the Catalan triangle and qualify as proper Riordan arrays for many choices of parameters. In particular, we use generalized path pairs to provide a new combinatorial interpretation for all entries in every proper Riordan array $\\mathcal{R}(d(t),h(t))$ of the form $d(t) = C_k(t)^i$, $h(t) = C_k(t)^k$, where $1 \\leq i \\leq k$ and $C_k(t)$ is the generating function for some sequence of Fuss-Catalan numbers (some $k \\geq 2$). Closed formulas are then provided for the number of generalized path pairs across an even broader range of parameters, as well as for the number of weak path pairs with a fixed number of non-initial intersections.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-03T18:13:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "generalized path pairs",
      "fuss-catalan triangles",
      "proper riordan array",
      "combinatorial interpretation",
      "enforce divisibility conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}