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

TL;DR

This paper investigates tiling properties of ribbon L-shaped n-ominoes for even n, revealing a rigidity property that simplifies tiling classification and introduces local move properties, with results not extending to odd n.

Contribution

It establishes a rigidity property for tilings with T_n when n is even, classifies rectangles that can be tiled, and demonstrates the local move property for these tilings.

Findings

Tiling of the first quadrant by T_n reduces to rectangles of size 2 x n or n x 2.

Rectangles are tileable by T_n if both sides are even and one is divisible by n.

The local move property holds for T_n with respect to specific rectangle tilings.

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.