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Signed Tilings by Ribbon L n-Ominoes, n Even, via Gröbner Bases 被引量:1
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作者 Kenneth Gill Viorel Nitica 《Open Journal of Discrete Mathematics》 2016年第3期185-206,共22页
Let T<sub>n </sub>be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T<sup>+</sup><sub>n</sub> be T<sub>n</sub> with an extra 2 x 2 square. We investiga... Let T<sub>n </sub>be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T<sup>+</sup><sub>n</sub> be T<sub>n</sub> with an extra 2 x 2 square. We investigate signed tilings of rectangles by T<sub>n</sub> and T<sup>+</sup><sub>n</sub> . We show that a rectangle has a signed tiling by T<sub>n</sub> 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<sup>+</sup><sub>n, </sub> 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 Gr&Ouml;bner 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. 展开更多
关键词 POLYOMINO replicating tile L-Shaped Polyomino Skewed L-Shaped Polyomino Signed Tilings Gröbner Basis Tiling Rectangles Coloring Invariants
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Signed Tilings by Ribbon L n-Ominoes, n Odd, via Gröbner Bases 被引量:1
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作者 Viorel Nitica 《Open Journal of Discrete Mathematics》 2016年第4期297-313,共17页
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. A consequence of our technique, based on the exhibition of an explicit Gr&#246;bner basis, is... 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. A consequence of our technique, based on the exhibition of an explicit Gr&#246;bner basis, is that any k-inflated copy of the skewed L n-omino has a signed tiling by skewed L n-ominoes. We also discuss regular tilings by ribbon L n-ominoes, n odd, for rectangles and more general regions. We show that in this case obstructions appear that are not detected by signed tilings. 展开更多
关键词 POLYOMINO replicating tile L-Shaped Polyomino Skewed L-Shaped Polyomino Signed Tilings Gröbner Basis Coloring Invariants
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