The Permanent is a function of a square matrix, a polynomial in the matrix elements. It's like the determinant, with N! terms, except that all the signs are +. [ A B ] Perm [ C D ] = AD+BC, while Det = AD-BC. It seems to be much harder to compute the permanent than the determinant. Some elementary row operations, like rowA += rowB, which preserve determinants, do not preserve permanents. These are what make the determinant easier to calculate. Permanents aren't nearly as useful as determinants, but they crop up in complexity theory. In particular, computing permanents can answer questions like "How many solutions are there to make this boolean formula true?" I have an "Is this known?" question for the group: Here's a theoretical way to compute the permanent of an NxN matrix M: Choose a random vector Z of N elements. Each entry is an independent randomly chosen complex number on the unit circle, |Zk|=1. Compute V = ZM, giving another vector of complex numbers. (The typical entry is unlikely to be on the unit circle.) Multiply together all the complex numbers in V, and divide by the product of the complex numbers in Z. In rough symbols, compute product(elements of V) ---------------------- = result P, another complex number. product(elements of Z) Then the expected value of P is the permanent of M. (I'm assuming the entries of Z are chosen uniformly.) [Is this already known?] This isn't a very practical way to compute permanents -- it bears a lot of similarity to Buffon's Needle method of computing pi. But the operation of choosing random elements on the unit circle suggests that it might be a good algorithm for quantum computing. Rich rcs@cs.arizona.edu