Symmetric polyomino tilings, tribones, ideals, and Groebner bases

Manuela Muzika Dizdarevic, Rade T. Zivaljevic

Published 2014-07-08Version 1

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.