A bijection proving the Aztec diamond theorem by combing lattice paths - Archive ouverte HAL Accéder directement au contenu
Rapport Année : 2012

A bijection proving the Aztec diamond theorem by combing lattice paths

Résumé

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.
Fichier principal
Vignette du fichier
Aztec.pdf (347.77 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-00734890 , version 1 (24-09-2012)

Identifiants

Citer

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

Altmetric

Partager

Gmail Facebook X LinkedIn More