  In 2003, Ciucu presented a unified way to enumerate tilings of lattice
regions by using a certain Reduction Theorem (Ciucu, Perfect Matchings and
Perfect Powers, Journal of Algebraic Combinatorics, 2003). In this paper we
continue this line of work by investigating new families of lattice regions
whose tilings are enumerated by perfect powers or products of several perfect
powers. We prove a multi-parameter generalization of Bo-Yin Yang's theorem on
fortresses (B.-Y. Yang, Ph.D. thesis, Department of Mathematics, MIT, MA,
1991). On the square lattice with zigzag paths, we consider two particular
families of regions whose numbers of tilings are always a power of 3 or twice a
power of 3. The latter result provides a new proof for a conjecture of Matt
Blum first proved by Ciucu. We also consider several new lattices obtained by
periodically applying two simple subgraph replacement rules to the square
lattice. On some of those lattices, we get new families of regions whose
numbers of tilings are given by products of several perfect powers. In
addition, we prove a simple product formula for the number of tilings of a
certain family of regions on a variant of the triangular lattice.
