  We introduce "fertility Wilf equivalence," "strong fertility Wilf
equivalence," and "postorder Wilf equivalence," three variants of Wilf
equivalence for permutation classes that formalize some phenomena that have
appeared in the study of West's stack-sorting map. We introduce "sliding
operators" and show that they induce useful bijections among sets of valid hook
configurations. Combining these maps with natural decompositions of valid hook
configurations, we give infinitely many examples of fertility, strong
fertility, and postorder Wilf equivalences. As a consequence, we obtain
infinitely many joint equidistribution results concerning many permutation
statistics. In one very special case, we reprove and extensively generalize a
result of Bouvel and Guibert. Another case reproves and generalizes a result of
the current author. A separate very special case proves and generalizes a
conjecture of the current author concerning stack-sorting preimages and the
Boolean-Catalan numbers. We end with two open questions.
