A Bijection Proving the Aztec Diamond Theorem by Combing Lattice Paths (original) (raw)

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Keywords: Aztec diamond, Domino tiling, Non-intersecting paths, Bijective proof, Algorithmic bijection

Abstract

We give a bijective proof of the Aztec diamond theorem, stating that there are 2n(n+1)/22^{n(n+1)/2}2n(n+1)/2 domino tilings of the Aztec diamond of order nnn. The proof in fact establishes a similar result for non-intersecting families of n+1n+1n+1 Schröder paths, with horizontal, diagonal or vertical steps, linking the grid points of two adjacent sides of an ntimesnn\times nntimesn 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 2n(n+1)/22^{n(n+1)/2}2n(n+1)/2 in number; it transforms them into non-intersecting families.

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How to Cite

Bosio, F., & van Leeuwen, M. A. A. (2013). A Bijection Proving the Aztec Diamond Theorem by Combing Lattice Paths. The Electronic Journal of Combinatorics, 20(4), #P24. https://doi.org/10.37236/2809