```-- | An interpretation of species as ordinary generating functions,
--   which count unlabelled structures.
module Math.Combinatorics.Species.Unlabelled
( unlabelled ) where

import Math.Combinatorics.Species.Types
import Math.Combinatorics.Species.Class
import Math.Combinatorics.Species.AST
import Math.Combinatorics.Species.CycleIndex

import qualified MathObj.PowerSeries as PS

import qualified Algebra.Differential as Differential

import NumericPrelude
import PreludeBase hiding (cycle)

needsCI :: String -> a
needsCI op = error ("unlabelled " ++ op ++ " must go via cycle index series.")

instance Differential.C GF where
differentiate = needsCI "differentiation"

instance Species GF where
singleton         = gfFromCoeffs [0,1]
set               = gfFromCoeffs (repeat 1)
cycle             = set
o                 = needsCI "composition"
cartesian         = needsCI "cartesian product"
fcomp             = needsCI "functor composition"
ofSize s p        = (liftGF . PS.lift1 \$ filterCoeffs p) s
ofSizeExactly s n = (liftGF . PS.lift1 \$ selectIndex n) s

unlabelledCoeffs :: GF -> [Integer]
unlabelledCoeffs (GF p) = PS.coeffs p ++ repeat 0

-- | 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
--   (@unlabelled s@ is guaranteed to be infinite).  For example:
--
-- > > 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 'unlabelled',
--   which is 'SpeciesAST' rather than the expected 'GF'.  The reason
--   is that although products and sums of unlabelled 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 'unlabelled' actually
--   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.
unlabelled :: SpeciesAST -> [Integer]
unlabelled s
| needsZ s  = unlabelledCoeffs . zToGF . reflect \$ s
| otherwise = unlabelledCoeffs . reflect \$ s
```