Aztec diamond

In combinatorial mathematics, an Aztec diamond of order n consists of all squares of a square lattice whose centers (x,y) satisfy |x| + |y| ≤ n. Here n is a fixed integer, and the square lattice consists of unit squares with the origin as a vertex of 4 of them, so that both x and y are half an odd integer.

The Aztec diamond theorem (Noam Elkies, Greg Kuperberg & Michael Larsen et al. 1992) states that the number of domino tilings of the Aztec diamond of order n is 2^n(n+1)/2. The arctic circle theorem says that a random tiling of a large Aztec diamond tends to be frozen outside a certain circle.

References

• Elkies, Noam; Kuperberg, Greg; Larsen, Michael; Propp, James (1992), "Alternating-sign matrices and domino tilings. I", Journal of Algebraic Combinatorics. An International Journal 1 (2): 111–132, doi:10.1023/A:1022420103267, ISSN 0925-9899, MR1226347 

• Stanley, Richard P. (1999), Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, ISBN 978-0-521-56069-6; 978-0-521-78987-5, MR1676282, http://www-math.mit.edu/~rstan/ec/ 
• Weisstein, Eric W., "Aztec Diamond" from MathWorld.