Signed tilings by ribbon L-shaped n-ominoes, n even, via Groebner bases
Viorel Niţică · arXiv (Cornell University) · 2016
We investigate signed tilings of rectangles by ribbon $L$-shaped $n$-ominoes, $n\ge 6$ even. We show that for $n=6$ a rectangle has a signed tiling by ribbon $L$-shaped hexominoes if and only if one of the sides of the rectangle is divisible by $6$. We show that a rectangle has a signed tiling by $\mathcal{T}_n$, $n\ge 8$ even, if and only if both sides of the rectangle are even and one of them is divisible by $n$, or if one of the sides is odd and the other side is divisible by $n\left (\frac{n}{2}-2\right ).$ Our proofs are based on the exhibition of explicit Gr\obner bases. In particular, this paper shows that some of the regular tiling results in \emph{ V.~Nitica, Every tiling of the first quadrant by ribbon $L$ $n$-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, {\em 5}, (2015) 11--25,} cannot be obtained from coloring invariants.