Double Aztec Rectangles

We investigate the connection between lozenge tilings and domino tilings by introducing a new family of regions obtained by attaching two different Aztec rectangles. We prove a simple product formula for the generating functions of the tilings of the new regions, which involves the statistics as in the Aztec diamond theorem (Elkies, Kuperberg, Larsen, and Propp, J. Algebraic Combin. 1992). Moreover, we consider the connection between the generating function and MacMahon's $q$-enumeration of plane partitions fitting in a given box