Experimenting with the Garsia–Milne Involution Principle

In 1981, Adriano Garsia and Steve Milne found the first bijective proof of the celebrated Rogers–Ramanujan identities. To achieve this feat, they invented a versatile tool that they called the Involution Principle. In this note we revisit this useful principle from a very general perspective, independent of its application to specific combinatorial identities, and will explore its complexity.