The Tilings of Deficient Squares by Ribbon L-Tetrominoes Are Diagonally Cracked
Abstract: We consider tilings of deficient rectangles by the set T4 of ribbon L-tetro-minoes. A tiling exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3). The majority of the tiles in a tiling follow the rectangular pattern, that is, are paired and each pair tiles a 2×4 rectangle. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The crack divides the square in two parts of equal area. The number of tilings of a (4m+1)×(4m+1) deficient square by T is equal to the number of tilings by dominoes of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T4  is twice the number of tilings by dominoes of a (2m+1)×(2m+1) deficient square, with the missing cell placed on the main diagonal. In both cases the counting is realized by an explicit function which is a bijection in the first case and a double cover in the second. If an extra 2×2 tile is added to T4 , we call the new tile set T+4. A tiling of a deficient rectangle by T+4 exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3). The majority of the tiles in a tiling follow the rectangular pattern, that is, are either paired tetrominoes and each pair tiles a 2×4 rectangle, or are 2×2 squares. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The number of tilings of a (4m+1)×(4m+1) deficient square by T+4 is greater than the number of tilings by dominoes and monomers of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T+4 is greater than twice the number of tilings by dominoes and monomers of a (2m+1)×(2m+1) deficient square, with the missing cell placed on the main diagonal. We also consider tilings by T4  and T+4 of other significant deficient regions. In particular we show that a deficient first quadrant, a deficient half strip, a deficient strip or a deficient bent strip cannot be tiled by T+4. Therefore T4  and T+4 give examples of tile sets that tile deficient rectangles but do not tile any deficient first quadrant, any deficient half strip, any deficient bent strip or any deficient strip.
Cite this paper: Nitica, V. (2017) The Tilings of Deficient Squares by Ribbon L-Tetrominoes Are Diagonally Cracked. Open Journal of Discrete Mathematics, 7, 165-176. doi: 10.4236/ojdm.2017.73015.
References

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

   Nitica, V. (2003) Rep-Tiles Revisited, in the Volume MASS Selecta: Teaching and Learning Advanced Undergraduate Mathematics. American Mathematical Society.

   Kasteleyn, P.M. (1961) The Statistics of Dimers on a Lattice. Physica, 27, 1209-1225.
https://doi.org/10.1016/0031-8914(61)90063-5

   Nitica, V. (2004-2005) Tiling a Deficient Rectangle by L-Tetrominoes. Journal of Recreational Mathematics, 33, 259-271.

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

   Hochberg, R. (2015) The Gap Number of the T-Tetromino. Discrete Mathematics, 338, 130-138.
https://doi.org/10.1016/j.disc.2014.09.001

   Zhan, S. (2012) Tiling a Deficient Rectangle with t-Tetrominoes.
https://www.math.psu.edu/mass/reu/2012/report/Tiling Deficient Rectangles with T-Tetrominoes.pdf

   Junius, P. and Nitica, V. (2017) Tiling Rectangles with Gaps by Ribbon Right Trominoes. Open Journal of Discrete Mathematics, 7, 87-102.
https://doi.org/10.4236/ojdm.2017.72010

   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.
https://doi.org/10.4236/ojdm.2015.52002

   Calderon, A., Fairchild, S., Nitica, V. and Simon, S. (2016) Tilings of Quadrants by L-Ominoes and Notched Rectangles. Topics in Recreational Mathematics, 7, 39-75.

   Nitica, V. (2016) On Tilings of Quadrants and Rectangles and Rectangular Pattern. Open Journal of Discrete Mathematics, 6, 252-271.
https://doi.org/10.4236/ojdm.2016.64028

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