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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References

  1. 1.
    R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1982.Google Scholar
  2. 2.
    E.F. Beckenbach, ed., Applied Combinatorial Mathematics, John Wiley, New York, 1964.Google Scholar
  3. 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. 4.
    C. Fan and F.Y. Wu, “General lattice model of phase transitions,” Phys. Rev. B2 (1970), 723–733.Google Scholar
  5. 5.
    J.A. Green, Polynomial Representations of GL n, Lecture Notes in Mathematics 830, Springer, Berlin, 1980.Google Scholar
  6. 6.
    W. Jockusch, “Perfect matchings and perfect squares,” Ph.D. thesis, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 1992.Google Scholar
  7. 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. 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. 9.
    E. Lieb, “Residual entropy of square ice,” Phys. Rev. 162 (1967), 162–172.Google Scholar
  10. 10.
    L. Lovász, Combinatorial Problems and Exercises, North-Holland, Amsterdam, 1979, prob. 4.29.Google Scholar
  11. 11.
    I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1979.Google Scholar
  12. 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. 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. 14.
    J.K. Percus, Combinatorial Methods, Courant Institute, New York, 1969.Google Scholar
  15. 15.
    G. Pólya and S. Szegö, Problems and Theorems in Analysis, Vol. II, Springer, New York, 1976, prob. 132, p. 134.Google Scholar
  16. 16.
    D.P. Robbins, “The story of 1, 2, 7, 42, 429, 7436,...,” Math. Intelligencer 13 (1991), 12–19.Google Scholar
  17. 17.
    D.P. Robbins and H. Rumsey, Jr., “Determinants and alternating sign matrices,” Adv. Math. 62 (1986), 169–184.Google Scholar
  18. 18.
    A.E. Spencer, “Problem E 2637,” Amer. Math. Monthly 84 (1977), 134–135; solution published in 85 (1978), 386-387.Google Scholar
  19. 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. 20.
    R. Stanley, Enumerative Combinatorics, Vol. I. Wadsworth and Brooks/Cole, Belmont, MA, 1986.Google Scholar
  21. 21.
    W. Thurston, “Conway's tiling groups,” Amer. Math. Monthly 97 (1990), 757–773.Google Scholar
  22. 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. 23.
    B.-Y. Yang, “Three enumeration problems concerning Aztec diamonds,” Ph.D. thesis, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 1991.Google Scholar

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

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