A combinatorial proof of Cayley's theorem on Pfaffians

Abstract

Cayley's theorem, published in 1847, asserts that any skew-symmetric determinant of even order is equal to the square of its Pfaffian. A proof of this result, depending on the combinatorial structures of determinants and Pfaffians, is presented here. It emerges that the theorem holds, essentially because the representation of a permutation of [1, 2,…, 2 m] as a product of disjoint cycles of odd parity relates it biuniquely to two independent partitions of the same set into pairs, while a permutation whose cyclic representation contains cycles of even parity corresponds to a term of the determinant which is eliminated by skew-symmetry.