Recent developments and open problems in the theory of permanents

Thus the definition of the permanent is similar to that of the determinant except for the sign associated with each term in the summation. This minor difference in the definition makes the two functions quite unlike each other. Perhaps the permanent cannot compete with its cousin, the determinant, in terms of the depth of theory and the breadth of applications, but it is safe to say that the permanent also exhibits both these characteristics in ample measure, a fact that has not received enough attention. The permanent has a rich structure when restricted to certain classes of matrices, particularly, matrices of zeros and ones, (entrywise) nonnegative matrices and positive semidefinite matrices. Furthermore, there is a certain similarity of its properties over the class of nonnegative matrices and the class of positive semidefinite matrices, which is not yet fully understood. In this article we describe some properties of permanents over these three classes, putting