A rigidity property of ribbon L-shaped n-ominoes and generalizations

Abstract

Let n integer greater or equal to 4 and even and let T_n be the set of ribbon L-shaped n-ominoes. We study tiling problems for regions in a square lattice by T_n. Our main result shows a remarkable rigidity property: a tiling of the first quadrant by T_n is possible if and only if it reduces to a tiling by 2 x n and n x 2 rectangles. An application is the classification of all rectangles that can be tiled by T_n: a rectangle can be tiled by T_n if and only if both of its sides are even and at least one side is divisible by n. Another application is the existence of the local move property for an infinite family of sets of tiles: T_n has the local move property for the class of rectangular regions with respect to the local moves that interchange a tiling of an n x n square by n/2 vertical rectangles, with a tiling by n/2 horizontal rectangles, each vertical/horizontal rectangle being covered by two ribbon L-shaped n-ominoes. We show that these results are not valid for any n odd. The rectangular pattern of a tiling persists if we add an extra 2 x 2 square to T_n. A rectangle can be tiled by the larger set of tiles if and only if it has both sides even. In contrast, the addition of an extra even x odd or odd x odd rectangle to one of the above sets of tiles allows for a tiling of the first quadrant that does not respect the rectangular pattern.