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A bijection proving the Aztec diamond theorem by combing lattice paths

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 $n+1$ 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 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.
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https://hal.archives-ouvertes.fr/hal-00734890
Contributor : Marc van Leeuwen Connect in order to contact the contributor
Submitted on : Monday, September 24, 2012 - 5:26:49 PM
Last modification on : Wednesday, October 20, 2021 - 3:22:17 AM
Long-term archiving on: : Tuesday, December 25, 2012 - 6:35:09 AM

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  • HAL Id : hal-00734890, version 1
  • ARXIV : 1209.5373

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Frédéric Bosio, Marc van Leeuwen. A bijection proving the Aztec diamond theorem by combing lattice paths. 2012. ⟨hal-00734890⟩

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