TITLE:
Signed Tilings by Ribbon L n-Ominoes, n Even, via Gröbner Bases
AUTHORS:
Kenneth Gill, Viorel Nitica
KEYWORDS:
Polyomino, Replicating Tile, L-Shaped Polyomino, Skewed L-Shaped Polyomino, Signed Tilings, Gröbner Basis, Tiling Rectangles, Coloring Invariants
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.6 No.3,
July
29,
2016
ABSTRACT: Let Tn be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T+n be Tn with an extra 2 x 2 square. We investigate signed tilings of rectangles by Tn and T+n . We show that a rectangle has a signed tiling by Tn 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 . We also show that a rectangle has a signed tiling by T+n, n≥6 even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by . Our proofs are based on the exhibition of explicit GrÖbner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in Nitica, V. (2015) Every tiling of the first quadrant by ribbon L n-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, 5, 11-25, cannot be obtained from coloring invariants.