{
  "annotation_id": "9576c4d0-fc15-4f26-a413-b5bf073b1e8a",
  "date_created": "2026-02-17T05:53:08.654000Z",
  "date_modified": "2026-02-17T05:53:08.654000Z",
  "file_hash": "3a185882e4c4bf2e764a45a5a537d885d7226ab68ab1866a23be95678c4a5e09",
  "private": false,
  "record": {
    "abstract": "In this paper, we study the extremal behaviour of deep holes in polyominoes. We determine the maximum number, $h_n$ of deep holes that an $n$-omino can enclose, ensuring that the boundary of each hole is disjoint from the boundaries of any other hole and from the outer boundary of the $n$-tile. Using the versatile application of Pick\u0027s theorem, we establish the lower and the upper bound for $h_n$, and show that $h_n=\\frac{n}{3}+o(n)$ asymptotically. To further develop these results, we compute $h_n$ as a function of $n$ for an infinite subset of positive integers.",
    "arxiv_id": "2601.06840",
    "authors": [
      "Djordje Baralic",
      "Shiven Uppal"
    ],
    "categories": [
      "math.CO",
      "math.OC"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "Polyominoes with maximal number of deep holes",
    "url": "https://arxiv.org/abs/2601.06840",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "b71417aa-08ac-4d7e-9f87-596c5e4abfc4",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}