  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.
