Enumeration of Certain Classes of Antichains

Goran Kilibarda

An antichain is here regarded as a hypergraph that satisfies the following property: neither of every two different edges is a subset of the other one. The paper is devoted to the enumeration of antichains given on an n-set and having one or more of the following properties: being labeled or unlabeled; being ordered or unordered; being a cover (or a proper cover); and finally, being a $T_0$-, $T_1$- or $T_2$-hypergraph. The problem of enumeration of these classes comprises, in fact, different modifications of Dedekind's problem. Here a theorem is proved, with the help of which a greater part of these classes can be enumerated. The use of the formula from the theorem is illustrated by enumeration of labeled antichains, labeled $T_0$-antichains, ordered unlabeled antichains, and ordered unlabeled $T_0$-antichains. Also a list of classes that can be enumerated in a similar way is given. Finally, we perform some concrete counting, and give a table of digraphs that we used in the counting process.