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Aztec Diamond


AztecDiamonds

An Aztec diamond of order n is the region obtained from four staircase shapes of height n by gluing them together along the straight edges. It can therefore be defined as the union of unit squares in the plane whose edges lie on the lines of a square grid and whose centers (x,y) satisfy

 |x-1/2|+|y-1/2|<=n.

The first few are illustrated above. The number of squares in the Aztec diamond of order n is 2n(n+1), giving for n=1, 2, ... the values 4, 12, 24, 40, 60, ... (OEIS A046092).

The number of domino tilings of an order n Aztec diamond is 2^(T_n), where T_n is the triangular number n(n+1)/2 (Elkies et al. 1992).

Note that Wassermann appears to use an incorrect definition of the Aztec diamond that is actually equivalent to that of a centered square number.


See also

Centered Square Number, Diamond, Triangular Number

Portions of this entry contributed by Gary Russell

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References

Andrews, G. E. and Eriksson, K. Integer Partitions, 2nd rev. ed. New York: Cambridge University Press, 2004.Elkies, N.; Kuperberg, G.; Larsen, M.; and Propp, J. "Alternating-Sign Matrices and Domino Tilings." J. of Alg. Comb. 1, 111-132, 1992.Edu, S.-P. and Fu, T.-S. "A Simple Proof of the Aztec Diamond Theorem." 2 Dec 2004. http://arxiv.org/abs/math.CO/0412041.Sloane, N. J. A. Sequence A046092 in "The On-Line Encyclopedia of Integer Sequences."Wassermann, A. "Covering the Aztec Diamond with One-Sided Tetrasticks." http://did.mat.uni-bayreuth.de/wassermann/tetrastick.pdf.

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Aztec Diamond

Cite this as:

Russell, Gary and Weisstein, Eric W. "Aztec Diamond." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AztecDiamond.html

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