We fix n and say a square in the two-dimensional grid indexed by (x,y)
has color c if x+y≡c(modn). A {\it 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 {\it 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 (in the area of R) algorithm for
determining whether R can be tiled with ribbon tiles of order n and
producing such a tiling when one exists. We also 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. Some of our results are generalizations of
known results for order-2 ribbon tilings (a.k.a. domino tilings). We also
discuss applications of multidimensional height functions to a broader class of
polyomino tiling problems.Comment: 25 pages, 7 figures. This version has been slightly revised (new
references, a new illustration, and a few cosmetic changes). To appear in
Transactions of the American Mathematical Societ