Lexicographic Enumeration of Set Partitions

TL;DR

This paper reviews lexicographic algorithms for enumerating set partitions, emphasizing their simplicity, efficiency, and educational value, without introducing new methods.

Contribution

It provides a clear explanation and implementation guidance for existing set partition enumeration algorithms, focusing on their practical and educational aspects.

Findings

Algorithms operate in linear space

Enumeration advances in constant amortized time

Methods are easy to implement in high-level languages

Abstract

In this report, we summarize the set partition enumeration problems and thoroughly explain the algorithms used to solve them. These algorithms iterate through the partitions in lexicographic order and are easy to understand and implement in modern high-level programming languages, without recursive structures and jump logic. We show that they require linear space in respect to the set cardinality and advance the enumeration in constant amortized time. The methods discussed in this document are not novel. Our goal is to demonstrate the process of enumerating set partitions and highlight the ideas behind it. This work is an aid for learners approaching this enumeration problem and programmers undertaking the task of implementing it.