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Signed Tilings by Ribbon l n-Ominoes, n Odd, via Gröbner Bases

Open Journal of Discrete Mathematics 2016年第4期6卷 297-313页

作者： Viorel Nitica Department of Mathematics West Chester University West Chester PA USA

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 Gröbner 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.

关键词： polyomino Replicating Tile l-shaped polyomino Skewed l-shaped polyomino Signed Tilings Gröbner Basis Coloring Invariants

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

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Open Journal of Discrete Mathematics 2016年第3期6卷 185-206页

作者： Kenneth Gill Viorel Nitica Department of Mathematics West Chester University West Chester USA

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.

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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On Tilings of Quadrants and Rectangles and Rectangular Pattern

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Open Journal of Discrete Mathematics 2016年第4期6卷 351-371页

作者： Viorel Nitica Department of Mathematics West Chester University of Pennsylvania West Chester PA USA

The problem of tiling rectangles by polyominoes generated large interest. A related one is the problem of tiling parallelograms by twisted polyominoes. Both problems are related with tilings of (skewed) quadrants by polyominoes. Indeed, if all tilings of a (skewed) quadrant by a tile set can be reduced to a tiling by congruent rectangles (parallelograms), this provides information about tilings of rectangles (parallelograms). We consider a class of tile sets in a square lattice appearing from arbitrary dissections of rectangles in two l-shaped polyominoes and from symmetries of these tiles about the first bisector. Only translations of the tiles are allowed in a tiling. If the sides of the dissected rectangle are coprime, we show the existence of tilings of all (skewed) quadrants that do not follow the rectangular (parallelogram) pattern. If one of the sides of the dissected rectangle is 2 and the other is odd, we also show tilings of rectangles by the tile set that do not follow the rectangular pattern. If one of the sides of the dissected rectangle is 2 and the other side is even, we show a new infinite family of tile sets that follows the rectangular pattern when tiling one of the quadrants. For this type of dis-section, we also show a new infinite family that does not follow the rectangular pattern when tiling rectangles. Finally, we investigate more general dissections of rectangles, with. Here we show infinite families of tile sets that follow the rectangular pattern for a quadrant and infinite families that do not follow the rectangular pattern for any quadrant. We also show, for infinite families of tile sets of this type, tilings of rectangles that do not follow the rectangular pattern.

关键词： polyomino l-shaped polyomino Skewed l-shaped polyomino Tiling Rectangles Tiling Quadrants Tiling Parallelograms Rectangular Pattern for Tiling Quadrants/Rectangles

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