An Exploration of the Permanent-Determinant Method

The permanent-determinant method and its generalization, the HafnianPfaffian method, are methods to enumerate perfect matchings of plane graphs that were 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 × b × c plane partitions all have the same cokernel. 4. Two symmetry classes of plane partitions can be enumerated with almost no calculation. Submitted: October 16, 1998; Accepted: November 9, 1998 [Also available as math.CO/9810091] The permanent-determinant method and its generalization, the Hafnian-Pfaffian method, is a method to enumerate perfect matchings of plane graphs that was discovered by P. W. Kasteleyn [18]. Given a bipartite plane graph Z, the method produces a matrix whose determinant is the number of perfect matchings of Z. Given a non-bipartite plane graph Z, it produces a Pfaffian with the same property. The method can be used to enumerate symmetry classes of plane partitions [21, 22] and domino tilings of an Aztec diamond [45] and is related to some recent factorizations of the number of matchings of plane graphs with symmetry [5, 15]. It is related to