Enumeration in Classes of Ordered Structures

Let K be a class of finite structures. There are two obvious enumeration questions to ask of K: what is the set of cardinalities of members of K (the spectrum problem) and, for each n ≥ 1 what is the number of pairwise non-isomorphic members of K of cardinality n (the fine spectrum problem). If we restrict our attention to classes of partially ordered structures, the spectrum problem is usually trivial since the standard classes considered usually contain the class of chains, while the fine spectrum problem is usually hopeless since it seems impossible to enumerate ordered sets much more complicated than chains or antichains.