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-integers.^[1]

The Aztec diamond theorem states that the number of domino tilings of the Aztec diamond of order n is 2^n(n+1)/2.^[2] The arctic circle theorem says that a random tiling of a large Aztec diamond tends to be frozen outside a certain circle.^[3]

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  An Aztec diamond of order 4, with 1024 domino tilings

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  One possible tiling.

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  A tiling of a hexagon chosen uniformly at random, with the "frozen" tiles being depicted in white. (Arctic Circle theorem.)

It is common to color the tiles in the following fashion. First consider a checkerboard coloring of the diamond. Each tile will cover exactly one black square. Vertical tiles where the top square covers a black square, is colored in one color, and the other vertical tiles in a second. Similarly for horizontal tiles.

References

1. Stanley, Richard P. (1999), Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, ISBN 978-0-521-56069-6, MR 1676282
2. 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, MR 1226347
3. Jockusch, William; Propp, James; Shor, Peter (1998), Random Domino Tilings and the Arctic Circle Theorem

External links

• Weisstein, Eric W., "Aztec Diamond", MathWorld.