Computations versus bijections for tiling enumeration

Abstract

The number of domino tilings of an Aztec rectangle is known to be 2(n+12)∏1≤i<j≤nkj−kij−i, while the number of lozenge tilings of a trapezoid is known to be ∏1≤i<j≤nkj−kij−i, where k1<k2<…<kn prescribes the positions of certain defects along one side of the rectangle or trapezoid, respectively. It is shown that these objects can naturally be extended to all (k1,…,kn)∈Zn in such a way that the signed enumeration of the extended objects is given by the very same formula as the (restricted) straight enumeration. The main purpose of this article is to provide first combinatorial proofs of these facts. These proofs are derived from “computational” proofs, but we seek to compare them to known combinatorial constructions whenever possible. This reveals among other things that we have constructed an extension of urban renewal. This extension also played (in disguised form) a fundamental role in the recent first bijective proof of the alternating sign matrix theorem of Konvalinka and the author, and one important motivation for the results presented in this paper is to work towards a significant simplification of this proof to the effect that it has a more combinatorial and less computational flavor.