Symmetric polyomino tilings, tribones, ideals, and Gröbner bases | Zendy

Manuela Muzika-Dizdarevic | Zendy; Rade T. Živaljević | Zendy

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Symmetric polyomino tilings, tribones, ideals, and Gröbner bases

Author(s) -

Manuela Muzika-Dizdarevic,

Rade T. Živaljević

Publication year - 2015

Publication title -

publications de l institut mathematique

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.246

H-Index - 17

eISSN - 1820-7405

pISSN - 0350-1302

DOI - 10.2298/pim1512001m

Subject(s) - polyomino , combinatorics , mathematics , substitution tiling , planar , hexagonal tiling , integer (computer science) , lattice (music) , hexagonal lattice , class (philosophy) , discrete mathematics , geometry , computer science , physics , computer graphics (images) , condensed matter physics , regular polygon , antiferromagnetism , acoustics , programming language , grid , artificial intelligence

We apply the theory of Groebner 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 signed tiling by three-in-line polyominoes (tribones) symmetric with respect to the 120 degrees rotation of the triangle if and only if either N=27r-1 or N=27r for some integer r.

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