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DOI: 10.4230/LIPIcs.ISAAC.2017.5

On the Number of p4-Tilings by an n-Omino

Authors: Kazuyuki Amano and Yoshinobu Haruyama

Published in: LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

Abstract

A plane tiling by the copies of a polyomino is called isohedral if every pair of copies in the tiling has a symmetry of the tiling that maps one copy to the other. We show that, for every $n$-omino (i.e., polyomino consisting of n cells), the number of non-equivalent isohedral tilings generated by 90 degree rotations, so called p4-tilings or quarter-turn tilings, is bounded by a constant (independent of n). The proof relies on the analysis of the factorization of the boundary word of a polyomino.

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Kazuyuki Amano and Yoshinobu Haruyama. On the Number of p4-Tilings by an n-Omino. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 5:1-5:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{amano_et_al:LIPIcs.ISAAC.2017.5,
  author =	{Amano, Kazuyuki and Haruyama, Yoshinobu},
  title =	{{On the Number of p4-Tilings by an n-Omino}},
  booktitle =	{28th International Symposium on Algorithms and Computation (ISAAC 2017)},
  pages =	{5:1--5:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-054-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{92},
  editor =	{Okamoto, Yoshio and Tokuyama, Takeshi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.5},
  URN =		{urn:nbn:de:0030-drops-82498},
  doi =		{10.4230/LIPIcs.ISAAC.2017.5},
  annote =	{Keywords: polyomino, plane tiling, isohedral tiling, word factorization}
}

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On the Number of p4-Tilings by an n-Omino

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