A bijection proving the Aztec diamond theorem by combing lattice paths

作者：Bosio Frederic^*; van Leeuwen Marc A A

来源：Electronic Journal of Combinatorics, 2013, 20(4): P24.

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 Schroder paths, with horizontal, diagonal or vertical steps, linking the grid points of two adjacent sides of an n x 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 %26quot;combing%26quot; 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.

• 出版日期2013-11-29