Let P be a finite partially ordered set, then an antichain in P is a set of pairwise incomparable elements. Antichains are also called Sperner systems in older literature (Comtet 1974).

For example, consider P to be a family of subsets together with the subset relation (i.e., s_1<=s_2 if s_1 is a subset of s_2). The following table gives the antichains on the set of subsets (i.e., the power set) of the n-set {1,2,3,...,n} for small n.


The number of antichains on the n-set {1,2,...,n} for n=0, 1, 2, ..., are 1, 2, 5, 19, 167, ... (OEIS A014466). If the empty set is not considered a valid antichain, then these reduce to 0, 1, 4, 18, 166, ... (OEIS A007153; Comtet 1974, p. 273).

The number of antichains on the n-set are equal to the number of monotonic increasing Boolean functions of n variables, and also the number of free distributive lattices with n generators (Comtet 1974, p. 273). Determining these numbers is known as Dedekind's problem, and their values for n=0, 1, ... are called Dedekind numbers (Jäkel 2023), though they are conventionally defined as the numbers obtained by adding one to OEIS A014466, i.e., 2, 3, 6, 20, 168, 7581, 7828354, ... (OEIS A000372; Speciner 1972).

The partial order width of P is the maximum cardinal number of an antichain in P. For a partial order, the size of the longest antichain is called the partial order width w(P). Sperner (1928) proved that the maximum size (and hence the width of the partial order) of an antichain containing n elements is

 w_(max(n))=(n; |_n/2_|),

where (n; k) is a binomial coefficient and |_n_| is the floor function.

See also

Boolean Function, Chain, Dedekind Number, Dedekind's Problem, Dilworth's Lemma, Partial Order Width, Partially Ordered Set

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Weisstein, Eric W. "Antichain." From MathWorld--A Wolfram Web Resource.

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