Recently a method has been developed by Jen to enumerate limit cycles in cellular automata (CA) with periodic boundary conditions. This involves operations on a connectivity matrix whose elements are related to the invariance of a site in a particular neighborhood to application of the CA rule. We extend this method to the case of fixed boundary conditions, of interest in simulations. In this case, translational invariance is lost, and the enumeration procedure is much more tedious than with periodic boundary conditions. We show examples for a fixed-point, a period-two, and a period-three enumeration in considerable detail, and give results — in agreement with simulations — for the number of fixed points and period-two cycles in selected two-state, nearest-neighbor CA rules.