I agree that Joyal's species are sort of key here, although the kinds of data types being considered here are not exactly the same thing.

My guess is that 1/(1-A) should be seen as a kind of "generating function" for the formal power series 1 + A + A<sup>2</sup> + A<sup>3</sup> ... and that the relevance of power series can be explained by some simplicial structure hiding somewhere.  It's certainly true that one can take derivatives and integrals of these data types: see Conor McBride, *The Derivative of a Regular Type is its Type of One-Hole Contexts*.

There *is* a direct interpretation of division which comes up when computing "Taylor series" of these datatypes: e.g. the type A<sup>n</sup>/n! is the n-element multiset, since A<sup>n</sup> is an n-tuple and n! can be read as the action of the permutation group. Unfortunately, the formalism does not quite work out: see <http://www.cs.nott.ac.uk/~ctm/Dissect.pdf> for a reference.  This may be related to Baez and Dolan's work on groupoid cardinality (arxiv: math.QA/0004133).