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]
One of 1024 possible domino tilings of an order 4 Aztec diamondA domino tiling of an order-50 Aztec diamond, chosen uniformly at random. The four corners of the diamond outside of the roughly circular area are "frozen".
The Aztec diamond theorem states that the number of domino tilings of the Aztec diamond of order n is 2n(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]
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.
Knuth has also defined Aztec diamonds of order n + 1/2.[4] They are identical with the polyominoes associated with the centered square numbers.
^Stanley, Richard P. (1999), Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, ISBN 978-0-521-56069-6, MR 1676282, archived from the original on 2008-10-05, retrieved 2008-11-18
^Elkies, Noam; Kuperberg, Greg; Larsen, Michael; Propp, James (1992), "Alternating-sign matrices and domino tilings. I", Journal of Algebraic Combinatorics, 1 (2): 111–132, doi:10.1023/A:1022420103267, ISSN 0925-9899, MR 1226347
^Jockusch, William; Propp, James; Shor, Peter (1998), Random Domino Tilings and the Arctic Circle Theorem, arXiv:math/9801068, Bibcode:1998math......1068J
^Knuth, Donald E. (2019), "Pre-Fascicle 5c (section 7.2.2.1, Dancing Links)", The Art of Computer Programming, vol. 4, p. 93
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