Sum rules in multiphoton coincidence rates

We show that sums of carefully chosen coincidence rates in a multiphoton interferometry experiment can be simplified by replacing the original unitary scattering matrix with a coset matrix containing $0$s. The number and placement of these $0$s reduces the complexity of each term in the sum without affecting the original sum of rates. In particular, the evaluation of sums of modulus squared of permanents is shown to turn in some cases into a sum of modulus squared of determinants. The sums of rates are shown to be equivalent to the removal of some optical elements in the interferometer.