Summary: Two procedures to compute the output distribution \(^\varphi S\) of certain stack filters \(S\) (so called erosion-dilation cascades) are given. One rests on the disjunctive normal form of \(S\) and also yields the rank selection probabilities. The other is based on inclusion-exclusion and e.g. yields \(\varphi_S\) for some important \(LU\,LU\)-operators \(S\). Properties of \(\varphi_S\) can be used to characterize smoothing properties of \(S\). Also, in the same way as our polynomials \(\varphi_S\) are computed one could compute the reliability polynomial of a connected graph, or more generally the reliability polynomial w.r.t. any positive Boolean function.