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Alternating-Sign Matrices and Domino Tilings (Part I)

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.

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Elkies, N., Kuperberg, G., Larsen, M. et al. Alternating-Sign Matrices and Domino Tilings (Part I). Journal of Algebraic Combinatorics 1, 111–132 (1992). https://doi.org/10.1023/A:1022420103267

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  • DOI: https://doi.org/10.1023/A:1022420103267

  • tiling
  • domino
  • alternating-sign matrix
  • monotone triangle
  • representation
  • square ice