  The permanent-determinant method and its generalization, the Hafnian-Pfaffian
method, are methods to enumerate perfect matchings of plane graphs that was
discovered by P. W. Kasteleyn. We present several new techniques and arguments
related to the permanent-determinant with consequences in enumerative
combinatorics. Here are some of the results that follow from these techniques:
  1. If a bipartite graph on the sphere with 4n vertices is invariant under the
antipodal map, the number of matchings is the square of the number of matchings
of the quotient graph.
  2. The number of matchings of the edge graph of a graph with vertices of
degree at most 3 is a power of 2.
  3. The three Carlitz matrices whose determinants count a x b x c plane
partitions all have the same cokernel.
  4. Two symmetry classes of plane partitions can be enumerated with almost no
calculation.
