An interpretation of species as ordinary generating functions, which count unlabeled structures.
Extract the coefficients of an ordinary generating function as a
list of Integers. In particular,
number of unlabeled
s-structures on an underlying set of size
unlabeled s is guaranteed to be infinite). For example:
> take 10 $ unlabeled octopi [0,1,2,3,5,7,13,19,35,59]
gives the number of unlabeled 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
SpeciesAST rather than the expected
GF. The reason
is that although products and sums of unlabeled species
correspond to products and sums of ordinary generating functions,
other operations such as 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
works by first reifying the species to an AST and checking which
operations are used in its definition, and then choosing to work
with cycle index series or directly with (much faster) ordinary
generating functions as appropriate.