Authors: Dizdarević, Manuela Muzika
Živaljević, Rade 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Symmetric polyomino tilings, tribones, ideals, and gröbner bases
Journal: Publications de l'Institut Mathematique
Volume: 98
Issue: 112
First page: 1
Last page: 23
Issue Date: 1-Jan-2015
Rank: M24
ISSN: 0350-1302
DOI: 10.2298/PIM1512001M
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 TN = T3k-1 and TN = T3k in a hexagonal lattice admit a signed tiling by three-in-line polyominoes (tribones) symmetric with respect to the 120° rotation of the triangle if and only if either N = 27r - 1 or N = 27r for some integer r ≥ 0. The method applied is quite general and can be adapted to a large class of symmetric tiling problems.
Keywords: Gröbner bases | Signed polyomino tilings | Tessellations
Publisher: Mathematical Institute of the SASA
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 
Topology, geometry and global analysis on manifolds and discrete structures 

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