Distances on lozenge tilings

Abstract

In this paper, a structural property of the set of lozenge tilings of a 2n-gon is highlighted. We introduce a simple combinatorial value called Hamming-distance, which is a lower bound for the the number of flips - a local transformation on tilings - necessary to link two tilings. We prove that the flip-distance between two tilings is equal to the Hamming-distance for n≤4. We also show, by providing a pair of so-called deficient tilings, that this does not hold for n≥6. We finally discuss the n∈=∈5 case, which remains open. © 2009 Springer Berlin Heidelberg.