Alternating sign matrices and domino tilings

TL;DR

This paper studies domino tilings of Aztec diamonds, providing a generating function that counts tilings, encodes domino orientations, and relates to alternating sign matrices and the square ice model.

Contribution

Introduces a generating function for domino tilings of Aztec diamonds, linking tiling enumeration with alternating sign matrices and ice models.

Findings

Derived a formula for the number of tilings of Aztec diamonds.

Connected domino tilings to alternating sign matrices.

Explored the orientation and accessibility of tilings.

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.