{
  "id": "2308.05167",
  "version": "v1",
  "published": "2023-08-09T18:11:15.000Z",
  "updated": "2023-08-09T18:11:15.000Z",
  "title": "Total positivity from a kind of lattice paths",
  "authors": [
    "Yu-Jie Cui",
    "Bao-Xuan Zhu"
  ],
  "categories": [
    "math.CO",
    "math.CA"
  ],
  "abstract": "Total positivity of matrices is deeply studied and plays an important role in various branches of mathematics. The main purpose of this paper is to study total positivity of a matrix $M=[M_{n,k}]_{n,k}$ generated by the weighted lattice paths in $\\mathbb{N}^2$ from the origin $(0,0)$ to the point $(k,n)$ consisting of types of steps: $(0,1)$ and $(1,t+i)$ for $0\\leq i\\leq \\ell$, where each step $(0,1)$ from height~$n-1$ gets the weight~$b_n(\\textbf{y})$ and each step $(1,t+i)$ from height~$n-t-i$ gets the weight $a_n^{(i)}(\\textbf{x})$. Using an algebraic method, we prove that the $\\textbf{x}$-total positivity of the weight matrix $[a_i^{(i-j)}(\\textbf{x})]_{i,j}$ implies that of $M$. Furthermore, using the Lindstr\\\"{o}m-Gessel-Viennot lemma, we obtain that both $M$ and the Toeplitz matrix of each row sequence of $M$ with $t\\geq1$ are $\\textbf{x}$-totally positive under the following three cases respectively: (1) $\\ell=1$, (2) $\\ell=2$ and restrictions for $a_n^{(i)}$, (3) general $\\ell$ and both $a^{(i)}_n$ and $b_n$ are independent of $n$. In addition, for the case (3), we show that the matrix $M$ is a Riordan array, present its explicit formula and prove total positivity of the Toeplitz matrix of the each column of $M$. In particular, from the results for Toeplitz-total positivity, we also obtain the P\\'olya frequency and log-concavity of the corresponding sequence. Finally, as applications, we in a unified manner establish total positivity and the Toeplitz-total positivity for many well-known combinatorial triangles, including the Pascal triangle, the Pascal square, the Delannoy triangle, the Delannoy square, the signless Stirling triangle of the first kind, the Legendre-Stirling triangle of the first kind, the Jacobi-Stirling triangle of the first kind, the Brenti's recursive matrix, and so on.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-09T18:11:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A20",
      "05A15",
      "15B05"
    ],
    "keywords": [
      "first kind",
      "toeplitz-total positivity",
      "toeplitz matrix",
      "study total positivity",
      "well-known combinatorial triangles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}