set of subsets of the n variables with set inclusion as the partial order. This construction produces the free distributive lattice with n generators. Thus, the Dedekind numbers count the number of elements in free distributive lattices. The Dedekind numbers also count (one more than) the number of abstract simplicial complexes on n elements, families of sets with the property that any subset of a set in the family also belongs to the family. Any antichain determines a simplicial complex, the family of subsets of antichain members, and conversely the maximal simplices in a complex form an antichain. Example For n = 2, there are six monotonic Boolean functions and six antichains of subsets of the two-element set {x,y}: The function f(x,y) = false that ignores its input values and always returns false corresponds to the empty antichain Ø. The logical conjunction f(x,y) = x ∧ y corresponds to the antichain { {x,y} } containing the single set {x,y}. The function f(x,y) = x that ignores its second argument and returns the first argument corresponds to the antichain { {x} } containing the single set {x} The function f(x,y) = y that ignores its first argument and returns the second argument corresponds to the antichain { {y} } containing the single set {y} The logical disjunction f(x,y) = x ∨ y corresponds to the antichain { {x}, {y} } containing the two sets {x} and {y}. The function f(x,y) = true that ignores its input values and always returns true corresponds to the antichain {Ø} containing only the empty set. Values The exact values of the Dedekind numbers are known for 0 ≤ n ≤ 8: 2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788 . The first six of these numbers are given by . M(6) was calculated by , M(7) was calculated by and , and M(8) by . If n is even, then M(n) must also be even. The calculation of the fifth Dedekind number M(5) = 7581 disproved a conjecture by Garrett Birkhoff that M(n) is always divisible by (2n − 1)(2n − 2). Summation formula rewrote the logical definition of antichains into the following arithmetic formula for the Dedekind numbers: where is the th bit of the number , which can be written using the floor function as However, this formula is not helpful for computing the values of M(n) for large n due to