An Algorithm to List All the Fixed-Point Free Involutions on a Finite Set

A fixed-point free involution on a finite set S is defined as a bijection I : S → S such as ∀e ∈ S, I(I(e)) = e and ∀e ∈ S, I(e) 6= e. In this article, the fixed-point free involutions are represented as partitions of the set S, and some properties linked to this representation are exhibited. Then an optimal algorithm to list all the fixed-point free involutions is presented. Its soundess relies on the representation of the fixed-point free involutions as partitions. Finally, an implementation of the algorithm is proposed, with an effective data representation.