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

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.