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Enumeration of certain classes of antichains

Kilibarda Goran (ALFA University, Department of Mathematics, Belgrade)

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 T0-, T1- or T2-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 T0-antichains, ordered unlabeled antichains, and ordered unlabeled T0-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.

Keywords: exact enumeration, monotone Boolean function, hypergraph, antichain, cover, bipartite graph, digraph, coloring of a digraph