Ars Combinatoria, 29 (1990), pp. 107-116.
Ömer Egecioglu
Skew Symmetric Matrices and the Pfaffian
Abstract.
The Pfaffian of the symbols a_ij, i< j has a combinatorial interpretation
as the signed weight generating function of perfect matchings in the complete graph.
By properly specializing the variables, this generating function reduces to the
signed weight generating function for the perfect matchings in an arbitrary simple
graph. We construct a weight and sign preserving bijection between two appropriately
constructed spaces of permutations: permutations with even cycles and pairs of
involutions without fixed points. This bijection gives a purely combinatorial proof
that the determinant of a zero axial skew-symmetric matrix is equal to the square
of the Pfaffian.
omer@cs.ucsb.edu