{
  "id": "2211.14909",
  "version": "v1",
  "published": "2022-11-27T18:16:40.000Z",
  "updated": "2022-11-27T18:16:40.000Z",
  "title": "An asymptotic lower bound on the number of polyominoes",
  "authors": [
    "Vuong Bui"
  ],
  "comment": "15 pages, 4 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $P(n)$ denote the number of polyominoes of $n$ cells, we show that there exist some positive numbers $A,T$ so that for every $n$, \\[ P(n) \\ge An^{-T\\log n} \\lambda^n, \\] where $\\lambda$ is Klarner's constant, that is $\\lambda=\\lim_{n\\to\\infty} \\sqrt[n]{P(n)}$. This is somewhat a step toward the well known conjecture that there exist positive $A,T$ so that $P(n)\\sim An^{-T}\\lambda^n$ for every $n$. Beside the above theoretical result, we also conjecture that the ratio of the number of inconstructible polyominoes over $P(n)$ is decreasing, by observing this behavior for the available values. The conjecture opens a nice approach to bounding $\\lambda$, since if it is the case, we can conclude that \\[ \\lambda < 4.1141, \\] which is quite close to the currently best lower bound $\\lambda > 4.0025$ and largely improves the currently best upper bound $\\lambda < 4.5252$. The approach is merely analytically manipulating the known or likely properties of the function $P(n)$, instead of giving new insights of the structure of polyominoes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-27T18:16:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotic lower bound",
      "polyominoes",
      "currently best upper bound",
      "currently best lower bound",
      "conjecture opens"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}