Enumeration of Certain Classes of $T_0$-hypergraphs

A hypergraph is a $T_0$-hypergraph if for every two different vertices of the hypergraph there exists an edge containing one of the vertices and not containing the other. A general method for the enumeration of certain classes of $T_0$-hypergraphs is given. $T_0$-hypergraphs that are considered here are singled out both by the properties they themselves satisfy and by the properties that dual hypergraphs associated with them satisfy. Though in case of the so-called ordered hipergraphs the property `to be a $T_0$-hypergraph' is reduced to the property `to having different columns' of corresponding matrices, combining this property with some properties, that we are considering here, gives sometimes classes of hypergraphs that are not so easy to enumerate. The problem of enumerating some of thus obtained classes remains unsolved. Special attention is devoted to enumerating of different classes of covers and connected hypergraphs.