  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.
