Skip to main content
Download PDF

Alternating-Sign Matrices and Domino Tilings (Part I)

Journal of Algebraic Combinatorics volume 1, pages 111–132 (1992)Cite this 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.

References

  1. R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1982.

    Google Scholar 

  2. E.F. Beckenbach, ed., Applied Combinatorial Mathematics, John Wiley, New York, 1964.

    Google Scholar 

  3. J.H. Conway and J.C. Lagarias, “Tilings with polyominoes and combinatorial group theory,” J. Combin. Theory Ser. A 53 (1990), 183–208.

    Google Scholar 

  4. C. Fan and F.Y. Wu, “General lattice model of phase transitions,” Phys. Rev. B2 (1970), 723–733.

    Google Scholar 

  5. J.A. Green, Polynomial Representations of GL n , Lecture Notes in Mathematics 830, Springer, Berlin, 1980.

    Google Scholar 

  6. W. Jockusch, “Perfect matchings and perfect squares,” Ph.D. thesis, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 1992.

  7. P.W. Kasteleyn, “The statistics of dimers on a lattice, I: the number of dimer arrangements on a quadratic lattice,” Physica 27 (1961), 1209–1225.

    Google Scholar 

  8. P.W. Kasteleyn, “Graph theory and crystal physics,” in Graph Theory and Theoretical Physics, F. Harary, ed., Academic Press, New York, 1967, pp. 43–110.

    Google Scholar 

  9. E. Lieb, “Residual entropy of square ice,” Phys. Rev. 162 (1967), 162–172.

    Google Scholar 

  10. L. Lovász, Combinatorial Problems and Exercises, North-Holland, Amsterdam, 1979, prob. 4.29.

    Google Scholar 

  11. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1979.

    Google Scholar 

  12. W.H. Mills, D.P. Robbins, and H. Rumsey, Jr., “Proof of the Macdonald conjecture,” Invent. Math. 66 (1982), 73–87.

    Google Scholar 

  13. W.H. Mills, D.P. Robbins, and H. Rumsey, Jr., “Alternating sign matrices and descending plane partitions,” J. Combin. Theory Ser. A 34 (1983), 340–359.

    Google Scholar 

  14. J.K. Percus, Combinatorial Methods, Courant Institute, New York, 1969.

    Google Scholar 

  15. G. Pólya and S. Szegö, Problems and Theorems in Analysis, Vol. II, Springer, New York, 1976, prob. 132, p. 134.

    Google Scholar 

  16. D.P. Robbins, “The story of 1, 2, 7, 42, 429, 7436,...,” Math. Intelligencer 13 (1991), 12–19.

    Google Scholar 

  17. D.P. Robbins and H. Rumsey, Jr., “Determinants and alternating sign matrices,” Adv. Math. 62 (1986), 169–184.

    Google Scholar 

  18. A.E. Spencer, “Problem E 2637,” Amer. Math. Monthly 84 (1977), 134–135; solution published in 85 (1978), 386-387.

    Google Scholar 

  19. R. Stanley, “A baker's dozen of conjectures concerning plane partitions,” in Combinatoire Énumérative, G. Labelle and P. Leroux, eds., Lecture Notes in Mathematics 1234, Springer-Verlag, Berlin, 1986, pp. 285–293.

    Google Scholar 

  20. R. Stanley, Enumerative Combinatorics, Vol. I. Wadsworth and Brooks/Cole, Belmont, MA, 1986.

    Google Scholar 

  21. W. Thurston, “Conway's tiling groups,” Amer. Math. Monthly 97 (1990), 757–773.

    Google Scholar 

  22. T. Tokuyama, “A generating function of strict Gelfand patterns and some formulas on characters of general linear groups,” J. Math. Soc. Japan 40 (1988), 671–685.

    Google Scholar 

  23. B.-Y. Yang, “Three enumeration problems concerning Aztec diamonds,” Ph.D. thesis, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 1991.

Download references

Author information

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
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Greg Kuperberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael Larsen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. James Propp
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

  • Issue Date:

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

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