  We introduce an algorithm that conjectures the structure of a permutation
class in the form of a disjoint cover of "rules"; similar to generalized grid
classes. The cover is usually easily verified by a human and translated into an
enumeration. The algorithm is successful on different inputs than other
algorithms and can succeed with any polynomial permutation class. We apply it
to every non-polynomial permutation class avoiding a set of length four
patterns. The structures found by the algorithm can sometimes allow an
enumeration of the permutation class with respect to permutation statistics, as
well as choosing a permutation uniformly at random from the permutation class.
We sketch a new algorithm formalizing the human verification of the conjectured
covers.
