Permanents through probability distributions

Abstract

We show that the permanent of a matrix can be written as the expectation value of a function of random variables each with zero mean and unit variance. This result is used to show that Glynn's theorem and a simplified MacMahon theorem extend from a common probabilistic interpretation of the permanent. Combining the methods in these two proofs, we prove a new result that relates the permanent of a matrix to the expectation value of a product of hyperbolic trigonometric functions, or, equivalently, the partition function of a spin system. We conclude by discussing how the main theorem can be generalized and how the techniques used to prove it can be applied to more general problems in combinatorics.