A Simple Proof of the Aztec Diamond Theorem

Abstract

Based on a bijection between domino tilings of an Aztec diamond and nonintersecting lattice paths, a simple proof of the Aztec diamond theorem is given in terms of Hankel determinants of the large and small Schröder numbers. MSC2000: 05A15

A generalization of Aztec diamond theorem, part II

“…Recall that the proof of Theorem 1.1 in this paper and the previous proof in [11] extend respectively the ideas of Kuo [10] and Fu and Eu [7] in the case of the Aztec diamonds. This suggests that Theorem 1.1 may also be proven by generalizing the other proofs of the Aztec diamond theorem (listed in the introduction).…”

Double Aztec rectangles

“…However, we can explain intuitively as follows. Similar to the correspondence of Fu and Eu in the case of Aztec diamonds [4], each domino tiling T of the double Aztec rectangle are in bijection with a family of nonintersecting lattice paths P T . Moreover, we will show that the difference between the rank of the tiling T and the total area underneath the lattice paths in the family P T is a constant, and that and the tiling T 0 corresponds to the family having the smallest underneath area.…”

State matrix recursion method and monomer–dimer problem

“…The Aztec diamond theorem from the excellent article of Elkies, Kuperberg, Larsen and Propp [1] states that the Aztec diamond of order n can be tiled by dominos in exactly 2 n(n+1)/2 ways. A simple proof of this theorem can be found in [2]. An augmented Aztec diamond of order n looks much like the Aztec diamond of order n, except that there are three long columns in the middle instead of two.…”