Publications de l'Institut Mathematique 2015 Volume 97, Issue 111, Pages: 69-87
https://doi.org/10.2298/PIM140406001K
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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