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

DOI: 10.1023/A:1022420103267

Cite this article as:
Elkies, N., Kuperberg, G., Larsen, M. et al. Journal of Algebraic Combinatorics (1992) 1: 111. doi:10.1023/A:1022420103267


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.

tilingdominoalternating-sign matrixmonotone trianglerepresentationsquare ice

Copyright information

© Kluwer Academic Publishers 1992

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
  3. 3.University of PennsylvaniaPhiladelphia
  4. 4.Massachusetts Institute of TechnologyCambridge