Alternating sign matrices and domino tilings

We introduce a family of planar regions, called Aztec diamonds, and study the ways in which these regions can be tiled by dominoes. Our main result is a generating function that not only gives the number of domino tilings of the Aztec diamond of order n but also provides information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from anothe r by means of local modifications. Several proofs of the formula are given. The problem turns out to have connections with the alternating sign matrices of Mills, Robbins, and Rumsey, as well as the square ice model studied by Lieb.

[1]  David P. Robbins,et al.  The Story of 1, 2, 7, 42, 429, 7436, … , 1991 .

[2]  W. Thurston.  Conway's tiling groups , 1990 .

[3]  Jeffrey C. Lagarias,et al.  Tiling with polyominoes and combinatorial group theory , 1990, J. Comb. Theory, Ser. A.

[4]  R. Baxter.  Solving Models in Statistical Mechanics , 1989 .

[5]  Takeshi Tokuyama.  A generating function of strict Gelfand patterns and some formulas on characters of general linear groups , 1988 .

[6]  Howard Rumsey,et al.  Determinants and alternating sign matrices , 1986 .

[7]  R. Stanley.  What Is Enumerative Combinatorics , 1986 .

[8]  Richard P. Stanley,et al.  A Baker's dozen of conjectures concerning plane partitions , 1986 .

[9]  David P. Robbins,et al.  Alternating Sign Matrices and Descending Plane Partitions , 1983, J. Comb. Theory, Ser. A.

[10]  G. Pólya,et al.  Problems and theorems in analysis , 1983 .

[11]  W. H. Mills,et al.  Proof of the Macdonald conjecture , 1982 .

[12]  R. Baxter.  Exactly solved models in statistical mechanics , 1982 .

[13]  James Green,et al.  Polynomial representations of GLn , 1980 .

[14]  L. Lovász.  Combinatorial problems and exercises , 1979 .

[15]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[16]  F. Y. Wu,et al.  General Lattice Model of Phase Transitions , 1970 .

[17]  G. Sposito,et al.  Graph theory and theoretical physics , 1969 .

[18]  Elliott H. Lleb.  Residual Entropy of Square Ice , 1967 .

[19]  E. F. Beckenbach,et al.  Applied Combinatorial Mathematics , 1965 .

[20]  P. W. Kasteleyn.  The Statistics of Dimers on a Lattice , 1961 .

[21]  P. W. Kasteleyn.  The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice , 1961 .