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From Aztec diamonds to pyramids: Steep tilings


Authors: Jérémie Bouttier, Guillaume Chapuy and Sylvie Corteel
Journal: Trans. Amer. Math. Soc. 369 (2017), 5921-5959
MSC (2010): Primary 05A17, 05A19, 05E05, 82B20
DOI: https://doi.org/10.1090/tran/7169
Published electronically: April 24, 2017
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Abstract: We introduce a family of domino tilings that includes tilings of the Aztec diamond and pyramid partitions as special cases. These tilings live in a strip of $ \mathbb{Z}^2$ of the form $ 1\leq x-y\leq 2\ell $ for some integer $ \ell \geq 1$, and are parametrized by a binary word $ w\in \{+,-\}^{2\ell }$ that encodes some periodicity conditions at infinity. Aztec diamond and pyramid partitions correspond respectively to $ w=(+-)^\ell $ and to the limit case $ w=+^\infty -^\infty $. For each word $ w$ and for different types of boundary conditions, we obtain a nice product formula for the generating function of the associated tilings with respect to the number of flips, that admits a natural multivariate generalization. The main tools are a bijective correspondence with sequences of interlaced partitions and the vertex operator formalism (which we slightly extend in order to handle Littlewood-type identities). In probabilistic terms our tilings map to Schur processes of different types (standard, Pfaffian and periodic). We also introduce a more general model that interpolates between domino tilings and plane partitions.


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Additional Information

Jérémie Bouttier
Affiliation: Institut de Physique Théorique, CEA, IPhT, 91191 Gif-sur-Yvette, France – and – CNRS URA 2306 and Département de Mathématiques et Applications, École normale supérieure, 45 rue d’Ulm, F-75231 Paris Cedex 05, France
Email: jeremie.bouttier@cea.fr

Guillaume Chapuy
Affiliation: LIAFA, CNRS et Université Paris Diderot, Case 7014, F-75205 Paris Cedex 13, France
Email: guillaume.chapuy@liafa.univ-paris-diderot.fr

Sylvie Corteel
Affiliation: LIAFA, CNRS et Université Paris Diderot, Case 7014, F-75205 Paris Cedex 13, France
Email: corteel@liafa.univ-paris-diderot.fr

DOI: https://doi.org/10.1090/tran/7169
Received by editor(s): July 16, 2014
Received by editor(s) in revised form: July 29, 2016
Published electronically: April 24, 2017
Additional Notes: All authors were partially funded by the Ville de Paris, projet Émergences Combinatoire à Paris
The first and second authors acknowledge partial support from Agence Nationale de la Recherche, grant number ANR 12-JS02-001-01 (Cartaplus)
The third author acknowledges support from Agence Nationale de la Recherche, grant number ANR-08-JCJC-0011 (ICOMB)
Article copyright: © Copyright 2017 American Mathematical Society