Ribbon Tilings and Multidimensional Height Functions

We fix n and say a square in the two-dimensional grid indexed by (x, y) has color c if x+ y ≡ c (mod n). A ribbon tile of order-n is a connected polyomino containing exactly one square of each color. We show that the set of order-n ribbon tilings of a simply connected region R is in one-to-one correspondence with a set of height functions from the vertices of R to Zn satisfying certain difference restrictions. It is also in one-to-one correspondence with the set of acyclic orientations of a certain partially oriented graph. Using these facts, we describe a linear-time algorithm for determining whether a given region can be tiled with ribbon tiles of order-n and resolve a conjecture of Pak by showing that any pair of order-n ribbon tilings of R can be connected by a sequence of local replacement moves. We also discuss applications of multidimensional height functions to a broader class of tiling