Counting transitive relations

In order to count partial orders on a set of n points, it seems necessary to explicitly construct a representative of every isomorphism type. While that is done, one might as well determine their automorphism groups. In this note it is shown how several other types of binary relations can be counted, based on an explicit enumeration of the partial orders and their automorphism groups. A partial order is a transitive, reflexive, and antisymmetric binary relation. Here we determine the number of quasi-orders q(n) (or finite topologies or transitive digraphs or reflexive transitive relations), the number of "soft" orders s(t) (or antisymmetric transitive relations), and the number of transitive relations t(n) on n points in terms of numbers of partial orders with a given automorphism group.