A Flexible Approach for the Enumeration of Down-Sets and its Application on Dedekind Numbers

TL;DR

This paper presents a flexible method for enumerating down-sets in finite posets, successfully computing Dedekind numbers up to b(6), demonstrating efficiency and scalability in combinatorial enumeration.

Contribution

The authors introduce a novel flexible approach for enumerating down-sets and apply it to compute Dedekind numbers b(5) and b(6), improving computational methods.

Findings

Successfully computed b(5) and b(6) Dedekind numbers

Developed two methods for b(5) calculation with different pre-calculation strategies

Evaluated 245 posets for b(6) calculation, demonstrating method scalability

Abstract

We introduce a flexible approach for the enumeration of the down-sets of a finite poset and test it with the calculation of the Dedekind numbers $b(5) = 7581$ and $b(6) = 7828354$. For the calculation of $b(5)$, we develop two methods of which the first one (without pre-calculations) requires simple evaluation of 80 posets and the second one (with pre-calculations) of 34 posets. The calculation of $b(6)$ (with pre-calculations) is done by evaluating 245 posets.