Dedekind number

In mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind, who defined them in 1897. The Dedekind number ${\displaystyle M(n)}$ is the number of monotone Boolean functions of ${\displaystyle n}$ variables. Equivalently, it is the number of antichains of subsets of an ${\displaystyle n}$-element set, the number of elements in a free distributive lattice with ${\displaystyle n}$ generators, and one more than the number of abstract simplicial complexes on a set with ${\displaystyle n}$ elements.

Accurate asymptotic estimates of ${\displaystyle M(n)}$ and an exact expression as a summation are known. However Dedekind's problem of computing the values of ${\displaystyle M(n)}$ remains difficult: no closed-form expression for ${\displaystyle M(n)}$ is known, and exact values of ${\displaystyle M(n)}$ have been found only for ${\displaystyle n\leq 9}$.