Back
 OJDM  Vol.6 No.4 , October 2016
Signed Tilings by Ribbon L n-Ominoes, n Odd, via Gröbner Bases
Abstract: 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.
Cite this paper: Nitica, V. (2016) Signed Tilings by Ribbon L n-Ominoes, n Odd, via Gröbner Bases. Open Journal of Discrete Mathematics, 6, 297-313. doi: 10.4236/ojdm.2016.64025.
References

[1]   Golomb, S.W. (1954) Checker Boards and Polyominoes. American Mathematical Monthly, 61, 675-682.
http://dx.doi.org/10.2307/2307321

[2]   Golomb, S.W. (1994) Polyominoes, Puzzles, Patterns, Problems, and Packings. Princeton University Press, Princeton.

[3]   Pak, I. (2000) Ribbon Tile Invariants. Transactions of the American Mathematical Society, 352, 5525-5561.
http://dx.doi.org/10.1090/S0002-9947-00-02666-0

[4]   Chao, M., Levenstein, D., Nitica, V. and Sharp, R. (2013) A Coloring Invariant for Ribbon L-Tetrominoes. Discrete Mathematics, 313, 611-621.
http://dx.doi.org/10.1016/j.disc.2012.12.007

[5]   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.
http://dx.doi.org/10.4236/ojdm.2015.52002

[6]   Golomb, S.W. (1964) Replicating Figures in the Plane. Mathematical Gazette, 48, 403-412.
http://dx.doi.org/10.2307/3611700

[7]   Nitica, V. (2003) Rep-Tiles Revisited. In: Katok, S., Sossinsky, A. and Tabachnikov, S., Eds., MASS Selecta: Teaching and Learning Advanced Undergraduate Mathematics, American Mathematical Society, Providence, 205-217.

[8]   Dizdarevic, M.M., Timotijevic, M. and Zivaljevic, R.T. (2016) Signed Polyomino Tilings by n-in-Line Polyominoes and Gröbner Bases. Publications de l’Institut Mathematique, Nouvelle série, 99, 31-42.

[9]   Barnes, F.W. (1982) Algebraic Theory of Brick Packing I. Discrete Mathematics, 42, 7-26.
http://dx.doi.org/10.1016/0012-365X(82)90049-8

[10]   Barnes, F.W. (1982) Algebraic Theory of Brick Packing II. Discrete Mathematics, 42, 129-144.
http://dx.doi.org/10.1016/0012-365X(82)90211-4

[11]   Gill, K. and Nitica, V. (2016) Signed Tilings by Ribbon L n-Ominoes, n Odd, via Gröbner bases. Open Journal of Discrete Mathematics, 7, 185-206.
http://dx.doi.org/10.4236/ojdm.2016.63017

[12]   Conway, J.H. and Lagarias, J.C. (1990) Tilings with Polyominoes and Combinatorial Group Theory. Journal Combinatorial Theory, Series A, 53, 183-208.
http://dx.doi.org/10.1016/0097-3165(90)90057-4

[13]   Bodini, O. and Nouvel, B. (2004) Z-Tilings of Polyominoes and Standard Basis. In Combinatorial Image Analysis, Springer, Berlin, 137-150.

[14]   Becker, T. and Weispfenning, V. (in cooperation with Heinz Krendel) (1993) Gröbner Bases. Springer-Verlag, Berlin.

[15]   Seidenberg, A. (1974) Constructions in Algebra. Transactions of the American Mathematical Society, 197, 273-313.

 
 
Top