From Aztec diamonds to pyramids: Steep tilings

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 Z 2 \mathbb {Z}^2 of the form 1 ≤ x − y ≤ 2 ℓ 1\leq x-y\leq 2\ell for some integer ℓ ≥ 1 \ell \geq 1 , and are parametrized by a binary word w ∈ { + , − } 2 ℓ w\in \{+,-\}^{2\ell } that encodes some periodicity conditions at infinity. Aztec diamond and pyramid partitions correspond respectively to w = ( + − ) ℓ w=(+-)^\ell and to the limit case w = + ∞ − ∞ w=+^\infty -^\infty . For each word w 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.