  We consider a generating function of the domino tilings of an Aztec rectangle
with several boundary unit squares removed. Our generating function involves
two statistics: the rank of the tiling and half number of vertical dominoes as
in the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp. In
addition, our work deduces a combinatorial explanation for an interesting
connection between the number of lozenge tilings of a semihexagon and the
number of domino tilings of an Aztec rectangle.
