Sets with involutions: Power group enumeration

The aim of this paper is a generalization of the well known classical power group enumeration by using sets with involutions instead of unstructured sets. The project to establish this was triggered by the rapid growth of topological enumeration, which used counting principles so far rather “ad hoc”, starting always from Cauchy–Frobenius’s lemma directly. In addition, using the new counting principles throws a new light on classical counting problems from graphical enumeration which have been solved so far with conventional methods.