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.
Highlights
We study tiling problems for regions in a square lattice by certain symmetries of an L-shaped polyomino
The L-shaped polyomino we study is placed in a square lattice and is made out of
Signed tilings by ribbon L n-ominoes, n odd are studied in [10], where we show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n
Summary
We study tiling problems for regions in a square lattice by certain symmetries of an L-shaped polyomino. The main result implies that a skewed L-shaped n-omino, n even, (see Figure 2(b)) is not a replicating tile of order k 2 for any odd k This development shows that the limitation of the orientations of the tiles can be of interest, in particular when investigating tiling problems in a skewed lattice. Theorem 1 shows that some tiling results for Tn , n ≥ 6 even, in [10] cannot be found via coloring arguments We recall that it is shown in [4] that a rectangle is signed tiled by T4 if and only if the sides are even and one side is divisible by 4. Guided by the work here, we conclude that Gröbner basis method for solving signed tiling problems with integer weights is sometimes more versatile and leads to stronger results Barnes method. For this particular family, we understand regular tilings of rectangles due to [5], but cannot decide if the results follow from coloring invariants
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