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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Contributeur : Marc Van Leeuwen <>
Soumis le : lundi 24 septembre 2012 - 17:26:49
Dernière modification le : vendredi 16 septembre 2016 - 15:16:28
Document(s) archivé(s) le : mardi 25 décembre 2012 - 06:35:09


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



Frédéric Bosio, Marc Van Leeuwen. A bijection proving the Aztec diamond theorem by combing lattice paths. 2012. <hal-00734890>



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