{
  "id": "2309.05903",
  "version": "v1",
  "published": "2023-09-12T01:24:55.000Z",
  "updated": "2023-09-12T01:24:55.000Z",
  "title": "Interlacing property of a family of generating polynomials over Dyck paths",
  "authors": [
    "Bo Wang",
    "Candice X. T. Zhang"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In the study of a tantalizing symmetry on Catalan objects, B\\'ona et al. introduced a family of polynomials $\\{W_{n,k}(x)\\}_{n\\geq k\\geq 0}$ defined by \\begin{align*} W_{n,k}(x)=\\sum_{m=0}^{k}w_{n,k,m}x^{m}, \\end{align*} where $w_{n,k,m}$ counts the number of Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$. They proposed two conjectures on the interlacing property of these polynomials, one of which states that $\\{W_{n,k}(x)\\}_{n\\geq k}$ is a Sturm sequence for any fixed $k\\geq 1$, and the other states that $\\{W_{n,k}(x)\\}_{1\\leq k\\leq n}$ is a Sturm-unimodal sequence for any fixed $n\\geq 1$. In this paper, we obtain certain recurrence relations for $W_{n,k}(x)$, and further confirm their conjectures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-12T01:24:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dyck paths",
      "interlacing property",
      "generating polynomials",
      "recurrence relations",
      "occurrences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}