{
  "abstract": "A de Bruijn polyomino has colored cells and includes exactly one instance of each possible coloring of another polyomino with those colors. This generalizes the notion of a de Bruijn sequence. We are interested in the de Bruijn polyominoes of minimum size. As a step toward finding these, we introduce the problem of finding the smallest polyominoes containing at least $N$ translated copies of the polyomino $p$, allowing overlaps. We say these are $(p,N)$-dense and have size $a_{p,N}$. For certain pairs of polyominoes $p$ and $q$ of equal size, we show there are polyomino transformations relating $a_{p,N}$ and $a_{q,N}$. Using these transformations, we identify classes of polyominoes which share the sequence $(a_{p,N})_{N=1}^\\infty$. We give closed forms for $(a_{p,N})_{N=1}^\\infty$ for most polyominoes with up to five cells. Leveraging these results we introduce novel de Bruijn polyominoes, as well as other colored polyforms with de Bruijn-like properties.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29231",
  "author_total": 4,
  "authors": [
    {
      "name": "D. M. Condon"
    },
    {
      "name": "Eli B. Dugan"
    },
    {
      "name": "Laney M. Goldman"
    },
    {
      "name": "Emily R. Williams"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2wb0ah0pz9gsb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29231",
  "fulltext_url": null,
  "id": "2wb0ah0pz9gsb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29231",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7204950432",
  "pdf_url": "https://arxiv.org/pdf/2608.29231",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2wb0ah0pz9gsb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29231",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29231"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Polyomino Density",
  "title_text": "Polyomino Density",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/2wb0ah0pz9gsb",
  "venue": "arXiv",
  "versions": []
}