Symmetric Polyomino Tilings, Tribones, Ideals, and Gröbner Bases

Manuela Muzika Dizdarević, Rade T. Živaljević

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.