Dedekind number

Summary

The Dedekind number $M(n)$ is the number of monotone Boolean functions of $n$ variables. It also counts the antichains of subsets of an $n$-element set. For $n \ge 1$, $M(n) - 2$ is the number of elements of the free distributive lattice on $n$ generators. The first values are $M(0) = 2$, $M(1) = 3$, $M(2) = 6$, $M(3) = 20$, $M(4) = 168$ and $M(5) = 7581$. The numbers grow very fast, and no closed formula for them is known; finding one is Dedekind's problem.