From Aztec diamonds to pyramids: Steep tilings

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 ℤ² of the form 1 ≤ x − y ≤ 2ℓ for some integer ℓ ≥ 1, and are parametrized by a binary word w ∈{+, −}^2ℓ that encodes some periodicity conditions at infinity. Aztec diamond and pyramid partitions correspond respectively to w =(+−)^ℓ and to the limit case w =+^∞−^∞. 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.