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 M(n) counts the number of monotonic Boolean functions of n variables. Equivalently, it counts the number of antichains of subsets of an n-element set, the number of elements in a free distributive lattice with n generators, or the number of abstract simplicial complexes with n elements.

Accurate asymptotic estimates of M(n) and an exact expression as a summation, are known. However Dedekind's problem of computing the values of M(n) remains difficult: no closed-form expression for M(n) is known, and exact values of M(n) have been found only for n ≤ 8.

A Boolean function is a function that takes as input n Boolean variables (that is, values that can be either false or true, or equivalently binary values that can be either 0 or 1), and produces as output another Boolean variable. It is monotonic if, for every combination of inputs, switching one of the inputs from false to true can only cause the output to switch from false to true and not from true to false. The Dedekind number M(n) is the number of different monotonic Boolean functions on n variables.

An antichain of sets (also known as a Sperner family) is a family of sets, none of which is contained in any other set. If V is a set of n Boolean variables, an antichain A of subsets of V defines a monotone Boolean function f, where the value of f is true for a given set of inputs if some subset of the true inputs to f belongs to A and false otherwise. Conversely every monotone Boolean function defines in this way an antichain, of the minimal subsets of Boolean variables that can force the function value to be true. Therefore, the Dedekind number M(n) equals the number of different antichains of subsets of an n-element set.

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