Journal of Algebraic Combinatorics

, Volume 1, Issue 2, pp 111–132

# Alternating-Sign Matrices and Domino Tilings (Part I)

• Noam Elkies
• Greg Kuperberg
• Michael Larsen
• James Propp
Article

## Abstract

We introduce a family of planar regions, called Aztec diamonds, and study tilings of these regions by dominoes. Our main result is that the Aztec diamond of order n has exactly 2n(n+1)/2 domino tilings. In this, the first half of a two-part paper, we give two proofs of this formula. The first proof exploits a connection between domino tilings and the alternating-sign matrices of Mills, Robbins, and Rumsey. In particular, a domino tiling of an Aztec diamond corresponds to a compatible pair of alternating-sign matrices. The second proof of our formula uses monotone triangles, which constitute another form taken by alternating-sign matrices; by assigning each monotone triangle a suitable weight, we can count domino tilings of an Aztec diamond.

tiling domino alternating-sign matrix monotone triangle representation square ice

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## Authors and Affiliations

• Noam Elkies
• 1
• Greg Kuperberg
• 2
• Michael Larsen
• 3
• James Propp
• 4
1. 1.Harvard UniversityCambridge
2. 2.University of California at BerkeleyBerkeley