Combinatorial Composition

Let and be Combinatorial Structures. We define their composition as:

with weight

Note that for this to be well defined, we need , otherwise there is infinitely many partitions of which contribute.

The above construction is naturally isomorphic to

where is the set of Partitions of .

Remark

This is not the same as:

See Unlabelled Composition. Ironically, the allegedly unlabelled composition builds -structures on ordered -tuples of -structures; while the labelled composition builds -structures on sets of -structures.