From Aztec diamonds to pyramids: Steep tilings

Bouttier, Jérémie; Chapuy, Guillaume; Corteel, Sylvie

doi:10.1090/tran/7169

From Aztec diamonds to pyramids: Steep tilings
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by Jérémie Bouttier, Guillaume Chapuy and Sylvie Corteel PDF

Trans. Amer. Math. Soc. 369 (2017), 5921-5959 Request permission

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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