A bijection proving the Aztec diamond theorem by combing lattice paths

TL;DR

This paper presents a bijective proof of the Aztec diamond theorem by constructing an invertible algorithm that transforms certain path families into non-intersecting ones, confirming the number of domino tilings.

Contribution

It introduces a novel bijective

Findings

Proves the number of tilings is 2^{n(n+1)/2}.

Establishes a bijection via a combing algorithm.

Links path families to domino tilings through a new method.

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 non-intersecting families of $n+1$ Schr\"oder 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 non-intersection 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 non-intersecting families.