[Paper Review] Alternating sign matrices and domino tilings

Noam D. Elkies, Greg Kuperberg+2 more|ArXiv.org|Jun 1, 1991

Advanced Combinatorial MathematicsMathematics18 references278 citations

TL;DR

This paper establishes a generating function for domino tilings of Aztec diamonds, revealing deep connections between tiling enumeration, alternating sign matrices, and the square ice model. It proves that the number of tilings of the order-$n$ Aztec diamond is $2^{n(n+1)/2}$, with a refined generating function $\mathrm{AD}(n;x,q) = \prod_{k=0}^{n-1}(1 + x q^{2k+1})^{n-k}$, which encodes vertical domino counts and tiling rank via local moves.

ABSTRACT

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 another 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.

Motivation & Objective

• To enumerate domino tilings of the Aztec diamond of order $n$ and refine this count using two statistics: number of vertical dominoes and tiling rank.
• To establish a bijection between tilings and bit-strings of length $n(n+1)/2$, linking combinatorial tiling structures to binary sequences.
• To demonstrate that the tiling rank—defined as the minimum number of local 90° rotations (elementary moves) needed to transform the all-horizontal tiling into a given tiling—coincides with a height-function-based invariant.
• To unify combinatorial objects including alternating sign matrices, monotone triangles, and square ice configurations via shared generating functions and order ideals.
• To explore connections between tiling enumeration and statistical mechanics, particularly the free-fermion case of the six-vertex (square ice) model with Aztec boundary conditions.

Proposed method

• Introduce the Aztec diamond as the region $\{(x,y) : |x| + |y| \leq n+1\}$ and define domino tilings as coverings by $1\times2$ or $2\times1$ tiles.
• Define the tiling rank via elementary moves: rotating a $2\times2$ block of two dominoes by 90°, and show that any tiling is reachable from the all-horizontal tiling through such moves.
• Construct a height function on the dual graph of the tiling using a checkerboard coloring and standard orientation, assigning values based on boundary conditions and edge directions.
• Use the height function to define an order ideal in a poset, which is then bijectively mapped to a tiling via stacked cubes, enabling enumeration.
• Establish a generating function $\mathrm{AD}(n;x,q) = \prod_{k=0}^{n-1}(1 + x q^{2k+1})^{n-k}$, where $x$ tracks vertical dominoes and $q$ tracks rank.
• Connect the tiling model to the six-vertex (square ice) model with Aztec boundary conditions, showing that the partition function equals $c^{n^2}$ under the condition $a^2 + b^2 = c^2$, corresponding to the free-fermion case.

Experimental results

Research questions

• RQ1 What is the exact number of domino tilings of the Aztec diamond of order $n$, and how can this be refined by statistics such as vertical domino count and tiling rank?
• RQ2 How does the tiling rank, defined via elementary moves, relate to a height function construction on the dual graph of the tiling?
• RQ3 What is the precise connection between domino tilings of Aztec diamonds and alternating sign matrices, and how does this link extend to monotone triangles and the six-vertex model?
• RQ4 Can the generating function for tilings be interpreted as a specialization of a larger multivariate generating function, and what does this imply for combinatorial symmetries?
• RQ5 How do Aztec boundary conditions in the square ice model affect the partition function and entropy compared to periodic boundary conditions?

Key findings

• The number of domino tilings of the Aztec diamond of order $n$ is exactly $2^{n(n+1)/2}$, a result confirmed through four distinct proofs.
• The refined generating function $\mathrm{AD}(n;x,q) = \prod_{k=0}^{n-1}(1 + x q^{2k+1})^{n-k}$ encodes both the number of vertical dominoes ($x$) and the tiling rank ($q$).
• The all-vertical tiling of the order-$n$ Aztec diamond has rank $n(n+1)(2n+1)/6$, the maximum possible among all tilings.
• There exists a bijection between domino tilings of the order-$n$ Aztec diamond and bit-strings of length $n(n+1)/2$, explaining the total count $2^{n(n+1)/2}$.
• The square ice model with Aztec boundary conditions and Boltzmann weights satisfying $a^2 + b^2 = c^2$ has a partition function equal to $c^{n^2}$, corresponding to the free-fermion case.
• The generating function $\mathrm{AD}(n;x,q)$ is a specialization of a larger $2n$-variable generating function, as shown via the shuffling method and further confirmed in subsequent work.