Publication : t14/095

From Aztec diamonds to pyramids: steep tilings

Bouttier J. (CEA, IPhT (Institut de Physique Théorique), F-91191 Gif-sur-Yvette, France)
Chapuy G. (LIAFA, CNRS et Université Paris Diderot, Case 7014, F-75205 Paris Cedex 13)
Corteel S. (LIAFA, CNRS et Université Paris Diderot, Case 7014, F-75205 Paris Cedex 13)
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 $1 leq x-y leq 2ell$ for some integer $ell geq 1$, and are parametrized by a binary word $win{+,-}^{2ell}$ 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.
Année de publication : 2017
Revue : Transactions of the American Mathematical Society 369 5921-5959 (2017)
DOI : 10.1090/tran/7169
Preprint : arXiv:1407.0665
Langue : Anglais

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