A Simple Proof for the Number of Tilings of Quartered Aztec Diamonds

In this paper a (lattice) region is a connected union of unit squares in the square lattice. A domino is the union of two unit squares that share an edge. A (domino) tiling of a region R is a covering of R by dominos such that there are no gaps or overlaps. Denote by T(R) the number of tilings of the region R. The Aztec diamond of order n is defined to be the union of all the unit squares with integral corners (x, y) satisfying |x|+ |y| 6 n+1. The Aztec diamond of order 8 is shown in Figure 1(a). In [3] it was shown that the number of tilings of the Aztec diamond of order n is 2. We are interested in three related families of regions first introduced by Jockusch and Propp [5] as follows. Divide the Aztec diamond of order n into two congruent parts by a zigzag cut with 2-unit steps (see Figure 1(b) for an example with n = 8). By superimposing two such zigzag cuts that pass the center of the Aztec diamond we partition the region into four parts, called quartered Aztec diamonds. Up to symmetry, there are essentially two different ways we can superimpose the two cuts. For one of them, we obtained a fourfold rotational symmetric pattern, and four resulting parts are congruent.