{
  "id": "2003.10098",
  "version": "v1",
  "published": "2020-03-23T06:49:36.000Z",
  "updated": "2020-03-23T06:49:36.000Z",
  "title": "Integer Sequences and Monomial Ideals",
  "authors": [
    "Chanchal Kumar",
    "Amit Roy"
  ],
  "comment": "21 pages, 3 figures. Comments are welcome",
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "abstract": "Let $\\mathfrak{S}_n$ be the set of all permutations of $[n]=\\{1,\\ldots,n\\}$ and let $W$ be the subset consisting of permutations $\\sigma \\in \\mathfrak{S}_n$ avoiding 132 and 312-patterns. The monomial ideal $I_W = \\left\\langle \\mathbf{x}^{\\sigma} = \\prod_{i=1}^n x_i^{\\sigma(i)} : \\sigma \\in W \\right\\rangle $ in the polynomial ring $R = k[x_1,\\ldots,x_n]$ over a field $k$ is called a hypercubic ideal in the article (Certain variants of multipermutohedron ideals, Proc. Indian Acad. Sci.(Math Sci. Vol. 126, No.4, (2016), 479-500). The Alexander dual $I_W^{[\\mathbf{n}]}$ of $I_W$ with respect to $\\mathbf{n}=(n,\\ldots,n)$ has the minimal cellular resolution supported on the first barycentric subdivision $\\mathbf{Bd}(\\Delta_{n-1})$ of an $n-1$-simplex $\\Delta_{n-1}$. We show that the number of standard monomials of the Artinian quotient $\\frac{R}{I_W^{[\\mathbf{n}]}}$ equals the number of rooted-labelled unimodal forests on the vertex set $[n]$. In other words, \\[ \\dim_k\\left(\\frac{R}{I_W^{[\\mathbf{n}]}}\\right) = \\sum_{r=1}^n r!~s(n,r) = {\\rm Per}\\left([m_{ij}]_{n \\times n} \\right),\\] where $s(n,r)$ is the (signless) Stirling number of the first kind and ${\\rm Per}([m_{ij}]_{n \\times n})$ is the permanent of the matrix $[m_{ij}]$ with $m_{ii}=i$ and $m_{ij}=1$ for $i \\ne j$. For various subsets $S$ of $\\mathfrak{S}_n$ consisting of permutations avoiding patterns, the corresponding integer sequences $\\left\\lbrace \\dim_k\\left(\\frac{R}{I_S^{[\\mathbf{n}]}}\\right) \\right\\rbrace_{n=1}^{\\infty}$ are identified.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-23T06:49:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13D02",
      "05E40"
    ],
    "keywords": [
      "monomial ideal",
      "integer sequences",
      "permutations",
      "first barycentric subdivision",
      "minimal cellular resolution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}