  Two permutation classes, the X-class and subpermutations of the increasing
oscillation are shown to exhibit an exponential Wilf-collapse. This means that
the number of distinct enumerations of principal subclasses of each of these
classes grows much more slowly than the class itself whereas a priori, based
only on symmetries of the class, there is no reason to expect this. The
underlying cause of the collapse in both cases is the ability to apply some
form of local symmetry which, combined with a greedy algorithm for detecting
patterns in these classes, yields a Wilf-collapse.
