EMIS ELibM Electronic Journals Publications de l'Institut Mathématique, Nouvelle Série
Vol. 98(112), pp. 1–23 (2015)

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Symmetric polyomino tilings, tribones, ideals, and Gröbner bases

Manuela Muzika Dizdarevic, Rade T. Zivaljevic

Faculty of Natural Sciences and Mathematics, Sarajevo, Bosnia and Herzegovina; Mathematical Institute SASA, Belgrade, Serbia

Abstract: We apply the theory of Gröbner bases to the study of signed, symmetric polyomino tilings of planar domains. Complementing the results of Conway and Lagarias we show that the triangular regions $T_N=T_{3k-1}$ and $T_N=T_{3k}$ in a hexagonal lattice admit a \emph{signed tiling} by three-in-line polyominoes (tribones) \emph{symmetric} with respect to the $120^{\circ}$ rotation of the triangle if and only if either $N=27r-1$ or $N=27r$ for some integer $r\geq 0$. The method applied is quite general and can be adapted to a large class of symmetric tiling problems.

Keywords: signed polyomino tilings; Gröbner bases; tessellations

Classification (MSC2000): 52C20; 13P10

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Electronic fulltext finalized on: 18 Nov 2015. This page was last modified: 6 Jan 2016.

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