
Frédéric Bosio

Marc A. A. van Leeuwen
Keywords:
Aztec diamond, Domino tiling, Nonintersecting paths, Bijective proof, Algorithmic bijection
Abstract
We give a bijective proof of the Aztec diamond theorem, stating that there are $2^{n(n+1)/2}$ domino tilings of the Aztec diamond of order $n$. The proof in fact establishes a similar result for nonintersecting families of $n+1$ Schröder paths, with horizontal, diagonal or vertical steps, linking the grid points of two adjacent sides of an $n\times n$ square grid; these families are well known to be in bijection with tilings of the Aztec diamond. Our bijection is produced by an invertible "combing'' algorithm, operating on families of paths without nonintersection condition, but instead with the requirement that any vertical steps come at the end of a path, and which are clearly $2^{n(n+1)/2}$ in number; it transforms them into nonintersecting families.