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

TL;DR

This paper introduces an efficient algorithm for enumerating all fixed-point free involutions on a finite set by leveraging their representation as partitions, with practical implementation details.

Contribution

The paper presents a novel optimal algorithm for listing all fixed-point free involutions based on their partition representation, including implementation insights.

Findings

Algorithm correctly lists all fixed-point free involutions

Representation as partitions simplifies the enumeration process

Implementation demonstrates practical efficiency

Abstract

An involution on a finite set is a bijection such as I(I(e))=e for all the element of the set. A fixed-point free involution on a finite set is an involution such as I(e)=e for none element of the set. In this article, the fixed-point free involutions are represented as partitions of the set and some properties linked to this representation are exhibited. Then an optimal algorithm to list all the fixed-point free involutions is presented. Its soundness 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.