Signed Tilings by Ribbon L n-Ominoes, n Even, via Gröbner Bases

Kenneth  Gill; Viorel  Nitica

doi:10.4236/ojdm.2016.63017

Kenneth Gill, Viorel Nitica

Abstract: Let T_nbe the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T⁺_n be T_n with an extra 2 x 2 square. We investigate signed tilings of rectangles by T_n and T⁺_n . We show that a rectangle has a signed tiling by T_n 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.

Keywords: Polyomino, Replicating Tile, L-Shaped Polyomino, Skewed L-Shaped Polyomino, Signed Tilings, Gröbner Basis, Tiling Rectangles, Coloring Invariants

Cite this paper: Gill, K. and Nitica, V. (2016) Signed Tilings by Ribbon L n-Ominoes, n Even, via Gröbner Bases. Open Journal of Discrete Mathematics, 6, 185-206. doi: 10.4236/ojdm.2016.63017.

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