The tilings of deficient squares by ribbon \(L\)-tetrominoes are diagonally cracked
Jan. 9, 2017
1:00 pm PHG 646
Abstract
We consider tilings of deficient rectangles by the set \(\mathcal{T}_4\) of
ribbon \(L\)-tetrominoes. 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 \((4m+1) \times (4m+1)\) and in an even
position if the square is \((4m+3)\times (4m+3)\). The majority of the
tiles in a tiling follow the rectangular pattern, that is, are paired and
each pair tiles a \(2\times 4\) 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 \(3\times 3\) 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 \((4m+1)\times (4m+1)\) deficient square by
\(\mathcal{T}_4\) is equal to the number of tilings by dominoes of a
\(2m\times 2m\) square. The number of tilings of a \((4m+3)\times (4m+3)\)
deficient square by \(\mathcal{T}_4\) is twice the number of tilings by
dominoes of a \((2m+1)\times (2m+1)\) 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. The crack in a square naturally propagates to a crack
in a larger square.
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