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PreprintCC BY-NC-ND 4.0arXiv

Polyomino Density

  • D. M. Condon
  • Eli B. Dugan
  • Laney M. Goldman
  • Emily R. Williams
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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 ap,N. For certain pairs of polyominoes p and q of equal size, we show there are polyomino transformations relating ap,N and aq,N. Using these transformations, we identify classes of polyominoes which share the sequence (ap,N)N=1∞. We give closed forms for (ap,N)N=1∞ 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.

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Cite

BibTeXRIS
BibTeX
@misc{condon2026polyomino,
  title = {{Polyomino Density}},
  author = {D. M. Condon and Eli B. Dugan and Laney M. Goldman and Emily R. Williams},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29231},
  eprint = {2608.29231},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29231},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Polyomino Density
AU  - D. M. Condon
AU  - Eli B. Dugan
AU  - Laney M. Goldman
AU  - Emily R. Williams
PY  - 2026
DA  - 2026/08/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.29231
UR  - https://arxiv.org/abs/2608.29231
AB  - 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^∞. We give closed forms for (a_p,N)_N=1^∞ 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.
ER  -