The Tilings of Deficient Squares by Ribbon <i>L</i>-Tetrominoes Are Diagonally Cracked

• 作者: Viorel Nitica
• 来源期刊: 离散数学期刊(英文)
• 学科: 医学
• 关键词: Tiling DEFICIENT RECTANGLES RIBBON Tetromino
• 年，卷（期）: 2017,（3）
• 页码范围: 165-176
• 页数: 12页
• 分类号: R73

摘要

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 T4? 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+<sub style="margin-left:-6px;">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 squar