{
  "id": "2004.00684",
  "version": "v1",
  "published": "2020-04-01T19:58:08.000Z",
  "updated": "2020-04-01T19:58:08.000Z",
  "title": "Combinatorics on lattice paths in strips",
  "authors": [
    "Nancy S. S. Gu",
    "Helmut Prodinger"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "For lattice paths in strips which begin at $(0,0)$ and have only up steps $U: (i,j) \\rightarrow (i+1,j+1)$ and down steps $D: (i,j)\\rightarrow (i+1,j-1)$, let $A_{n,k}$ denote the set of paths of length $n$ which start at $(0,0)$, end on heights $0$ or $-1$, and are contained in the strip $-\\lfloor\\frac{k+1}{2}\\rfloor \\leq y \\leq \\lfloor\\frac{k}{2}\\rfloor$ of width $k$, and let $B_{n,k}$ denote the set of paths of length $n$ which start at $(0,0)$ and are contained in the strip $0 \\leq y \\leq k$. We establish a bijection between $A_{n,k}$ and $B_{n,k}$. The generating functions for the subsets of these two sets are discussed as well. Furthermore, we provide another bijection between $A_{n,3}$ and $B_{n,3}$ by translating the paths to two types of trees.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-01T19:58:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19",
      "05C05"
    ],
    "keywords": [
      "lattice paths",
      "combinatorics",
      "generating functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}