{
  "id": "2012.14947",
  "version": "v1",
  "published": "2020-12-29T21:32:12.000Z",
  "updated": "2020-12-29T21:32:12.000Z",
  "title": "Colored Motzkin Paths of Higher Order",
  "authors": [
    "Isaac DeJager",
    "Madeleine Naquin",
    "Frank Seidl",
    "Paul Drube"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Motzkin paths of order-$\\ell$ are a generalization of Motzkin paths that use steps $U=(1,1)$, $L=(1,0)$, and $D_i=(1,-i)$ for every positive integer $i \\leq \\ell$. We further generalize order-$\\ell$ Motzkin paths by allowing for various coloring schemes on the edges of our paths. These $(\\vec{\\alpha},\\vec{\\beta})$-colored Motzkin paths may be enumerated via proper Riordan arrays, mimicking the techniques of Aigner in his treatment of Catalan-like numbers. After an investigation of their associated Riordan arrays, we develop bijections between $(\\vec{\\alpha},\\vec{\\beta})$-colored Motzkin paths and a variety of well-studied combinatorial objects. Specific coloring schemes $(\\vec{\\alpha},\\vec{\\beta})$ allow us to place $(\\vec{\\alpha},\\vec{\\beta})$-colored Motzkin paths in bijection with different subclasses of generalized $k$-Dyck paths, including $k$-Dyck paths that remain weakly above horizontal lines $y=-a$, $k$-Dyck paths whose peaks all have the same height modulo-$k$, and Fuss-Catalan generalizations of Fine paths. A general bijection is also developed between $(\\vec{\\alpha},\\vec{\\beta})$-colored Motzkin paths and certain subclasses of $k$-ary trees.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-29T21:32:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "colored motzkin paths",
      "higher order",
      "dyck paths",
      "proper riordan arrays",
      "coloring schemes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}