It prevents counting partitions, or to be a bit cheeky it's too fine grain of a partition of its .. uh is there a name for the range of a definition ?

For example the one element set should have 1 partition, under this definition there are as many partitions of a 1 element set as there are possible labels.

If I was going to define it I would say:

A labeled partition on S is a function P : S -> L, (L being the set of labels) whose range equals it's codomain. A unlabeled partition is a labeled partition where the labeles are the sets { x | x = P ...

Wait that has some circularity issues. Using an function to define its own codomain..