Permanents

T. S. Michael

The permanent of an n by n matrix A = [a_ij] is defined by

per(A) = sum a_{1, f(1)} * a_{2, f(2)} * ... * a_{n, f(n)}

where the sum extends over all n! permutations f of {1,2,...n}. This definition is identical to the definition of the determinant of A, except that there is no factor of +1 or -1 depending on the sign of the permutation f.

We discuss

• the combinatorial significance of the permanent
• some computational methods
• the proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix
• some published results of the speaker on permanents