The counting sequences of combinatorial classes are the main subject of study of enumerative combinatorics.
Two combinatorial classes are said to be isomorphic if they have the same numbers of objects of each size, or equivalently, if their counting sequences are the same.
[3] Frequently, once two combinatorial classes are known to be isomorphic, a bijective proof of this equivalence is sought; such a proof may be interpreted as showing that the objects in the two isomorphic classes are cryptomorphic to each other.
For instance, the triangulations of regular polygons (with size given by the number of sides of the polygon, and a fixed choice of polygon to triangulate for each size) and the set of unrooted binary plane trees (up to graph isomorphism, with a fixed ordering of the leaves, and with size given by the number of leaves) are both counted by the Catalan numbers, so they form isomorphic combinatorial classes.
[4] The theory of combinatorial species and its extension to analytic combinatorics provide a language for describing many important combinatorial classes, constructing new classes from combinations of previously defined ones, and automatically deriving their counting sequences.