Applicability of the Pfaffian Method to Combinatorial Problems on a Lattice

Abstract

For the solution of certain problems in statistical mechanics, a formulation in terms of the combinatorial problem of counting open or closed polygons drawn on a lattice has been found useful. The solution of such combinatorial problems has been given as a generating function which can be expressed as a Pfaffian. Necessary and sufficient conditions are given here to determine when such a Pfaffian representation is possible. If certain consistency conditions are satisfied, then the solution to the combinatorial problem can be written down immediately. The method makes some use of the ideas of emission and absorption operators for fermions.