{
  "id": "2106.05949",
  "version": "v1",
  "published": "2021-06-10T17:47:40.000Z",
  "updated": "2021-06-10T17:47:40.000Z",
  "title": "The lattice of arithmetic progressions",
  "authors": [
    "Marcel K. Goh",
    "Jad Hamdan"
  ],
  "comment": "8 pages, 1 figure, 2 tables",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "In this paper we investigate properties of the lattice $L_n$ of subsets of $[n] = \\{1,\\ldots,n\\}$ that are arithmetic progressions, under the inclusion order. For $n\\geq 4$, this poset is not graded and thus not semimodular. We start by deriving properties of the number $p_{nk}$ of arithmetic progressions of length $k$ in $[n]$. Next, we look at the set of chains in $L_n' = L_n\\setminus\\{\\emptyset,[n]\\}$ and study the order complex $\\Delta_n$ of $L_n'$. Third, we determine the set of coatoms in $L_n$ to give a general formula for the value of $\\mu_n$ evaluated at an arbitrary interval of $L_n$. In each of these three sections, we give an independent proof of the fact that for $n\\geq 2$, $\\mu_n(L_n) = \\mu(n-1)$, where $\\mu_n$ is the M\\\"obius function of $L_n$ and $\\mu$ is the classical (number-theoretic) M\\\"obius function. We conclude by computing the homology groups of $\\Delta_n$, providing yet another explanation for the formula of the M\\\"obius function of $L_n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-10T17:47:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A07"
    ],
    "keywords": [
      "arithmetic progressions",
      "homology groups",
      "properties",
      "general formula",
      "arbitrary interval"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}