{
  "id": "2305.16520",
  "version": "v1",
  "published": "2023-05-25T22:55:07.000Z",
  "updated": "2023-05-25T22:55:07.000Z",
  "title": "Note on the number of antichains in generalizations of the Boolean lattice",
  "authors": [
    "Jinyoung Park",
    "Michail Sarantis",
    "Prasad Tetali"
  ],
  "comment": "13 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "We give a short and self-contained argument that shows that, for any positive integers $t$ and $n$ with $t =O\\Bigl(\\frac{n}{\\log n}\\Bigr)$, the number $\\alpha([t]^n)$ of antichains of the poset $[t]^n$ is at most \\[\\exp_2\\Bigl(1+O\\Bigl(\\Bigl(\\frac{t\\log^3 n}{n}\\Bigr)^{1/2}\\Bigr)\\Bigr)N(t,n)\\,,\\] where $N(t,n)$ is the size of a largest level of $[t]^n$. This, in particular, says that if $t \\ll n/\\log^3 n$ as $n \\rightarrow \\infty$, then $\\log\\alpha([t]^n)=(1+o(1))N(t,n)$, giving a (partially) positive answer to a question of Moshkovitz and Shapira for $t, n$ in this range. Particularly for $t=3$, we prove a better upper bound: \\[\\log\\alpha([3]^n)\\le(1+4\\log 3/n)N(3,n),\\] which is the best known upper bound on the number of antichains of $[3]^n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-25T22:55:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boolean lattice",
      "antichains",
      "generalizations",
      "better upper bound",
      "largest level"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}