An interpretation of species as ordinary generating functions, which count unlabelled structures.
Extract the coefficients of an ordinary generating function as a
list of Integers. In particular,
unlabelled s !! n is the
number of unlabelled s-structures on an underlying set of size n.
> take 10 $ unlabelled octopi [0,1,2,3,5,7,13,19,35,59]
gives the number of unlabelled octopi on 0, 1, 2, 3, ... 9 elements.
Actually, the above is something of a white lie, as you may have
already realized by looking at the input type of
SpeciesAlg rather than the expected
reason is that although products and sums of unlabelled species
correspond to products and sums of ordinary generating functions,
composition and differentiation do not! In order to compute an
ordinary generating function for a species defined in terms of
composition and/or differentiation, we must compute the cycle
index series for the species and then convert it to an ordinary
generating function. So
unlabelled actually works by first
reifying the species to an AST and checking whether it uses
composition or differentiation, and using operations on cycle
index series if it does, and (much faster) operations directly on
ordinary generating functions otherwise.