Benchmarking of algorithms for set partitions

TL;DR

This paper reviews algorithms for generating set partitions, provides approximate formulas for counting them, and benchmarks various methods, recommending Djokic et al.'s algorithm for practical applications.

Contribution

It offers a comprehensive review and benchmarking of algorithms for set partition enumeration, including practical formulas and a recommended algorithm.

Findings

Djokic et al.'s algorithm performs best in benchmarks.

Approximate formulas effectively estimate the number of set partitions.

Benchmark results guide practical algorithm selection.

Abstract

Set partitions are arrangements of distinct objects into groups. The problem of listing all set partitions arises in a variety of settings, in particular in combinatorial optimization tasks. After a brief review, we give practical approximate formulas for determining the number of set partitions, both for small and large set sizes. Several algorithms for enumerating all set partitions are reviewed, and benchmarking tests were conducted. The algorithm of Djokic et al. is recommended for practical use.