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Alternating-Sign Matrices and Domino Tilings (Part II)

Journal of Algebraic Combinatorics volume 1, pages 219–234 (1992)Cite this article

Abstract

We continue the study of the family of planar regions dubbed Aztec diamonds in our earlier article and study the ways in which these regions can be tiled by dominoes. Two more proofs of the main formula are given. The first uses the representation theory of GL(n). The second is more combinatorial and produces a generating function that gives not only the number of domino tilings of the Aztec diamond of order n but also information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from another by means of local modifications. Lastly, we explore a connection between the combinatorial objects studied in this paper and the square-ice model studied by Lieb.

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Authors and Affiliations

  1. Harvard University, Cambridge, MA, 02138

    Noam Elkies

  2. University of California at Berkeley, Berkeley, CA, 94720

    Greg Kuperberg

  3. University of Pennsylvania, Philadelphia, PA, 19104

    Michael Larsen

  4. Massachusetts Institute of Technology, Cambridge, MA, 02139

    James Propp

Authors
  1. Noam Elkies
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  2. Greg Kuperberg
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  3. Michael Larsen
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  4. James Propp
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Elkies, N., Kuperberg, G., Larsen, M. et al. Alternating-Sign Matrices and Domino Tilings (Part II). Journal of Algebraic Combinatorics 1, 219–234 (1992). https://doi.org/10.1023/A:1022483817303

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  • DOI: https://doi.org/10.1023/A:1022483817303

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