  We consider several new families of graphs obtained from Aztec rectangle and
augmented Aztec rectangle graphs by trimming two opposite corners. We prove
that the perfect matchings of these new graphs are enumerated by powers of $2$,
$3$, $5$, and $11$. The result yields a proof of a conjectured posed by Ciucu.
In addition, we reveal a hidden relation between our graphs and the hexagonal
dungeons introduced by Blum.
