Enumeration of certain classes of antichains

Goran Kilibarda
2015 Publications de l'Institut Mathématique (Beograd)  
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
more » ... ct, 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. 2010 Mathematics Subject Classification: 05C30; 05C65.
doi:10.2298/pim140406001k fatcat:c2qckfz4l5espa74t4bmmmjxaa