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.