-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | Combinatorial species lite
--   
--   A simple library for combinatorial species with no dependencies but
--   base. See <a>http://github.com/akc/spe</a> for an introduction and
--   examples. If you want something more substantial, then you will most
--   likely be happier with the excellent species package by Brent Yorgey:
--   <a>http://hackage.haskell.org/package/species</a>
@package spe
@version 0.6


-- | License : BSD-3
--   
--   Species lite. See <a>http://github.com/akc/spe</a> for an introduction
--   and examples.
module Math.Spe

-- | A <a>combinatorial species</a> is an endofunctor on the category of
--   finite sets and bijections. We approximate this by a function as
--   defined.
type Spe a c = [a] -> [c]

-- | Species addition.
(.+.) :: Spe a b -> Spe a c -> Spe a (Either b c)

-- | The sum of a list of species of the same type.
assemble :: [Spe a c] -> Spe a c

-- | Species multiplication.
(.*.) :: Spe a b -> Spe a c -> Spe a (b, c)

-- | Ordinal L-species multiplication. Give that the underlying set is
--   sorted , elements in the left factor will be smaller than those in the
--   right factor.
(<*.) :: Spe a b -> Spe a c -> Spe a (b, c)

-- | The product of a list of species.
prod :: [Spe a b] -> Spe a [b]

-- | The ordinal product of a list of L-species.
ordProd :: [Spe a b] -> Spe a [b]

-- | The power F^k for species F.
(.^) :: Spe a b -> Int -> Spe a [b]

-- | The ordinal power F^k for L-species F.
(<^) :: Spe a b -> Int -> Spe a [b]

-- | The (partitional) composition F(G) of two species F and G. It is
--   usually used infix.
o :: Spe [a] b -> Spe a c -> Spe a (b, [c])

-- | The derivative d/dX F of a species F.
dx :: Spe (Maybe a) b -> Spe a b

-- | f <a>ofSize</a> n is like f on n element sets, but empty otherwise.
ofSize :: Spe a c -> Int -> Spe a c

-- | No structure on the empty set, but otherwise the same.
nonEmpty :: Spe a c -> Spe a c

-- | The species of sets.
set :: Spe a [a]

-- | The species characteristic of the empty set; the identity with respect
--   to species multiplication.
one :: Spe a ()

-- | The singleton species.
x :: Spe a a

-- | The species of ballots with k blocks
kBal :: Int -> Spe a [[a]]

-- | The species of ballots.
bal :: Spe a [[a]]

-- | The species of set partitions.
par :: Spe a [[a]]

-- | The species of lists (linear orders) with k elements.
kList :: Int -> Spe a [a]

-- | The species of lists (linear orders)
list :: Spe a [a]

-- | The species of cycles.
cyc :: Spe a [a]

-- | The species of permutations (sets of cycles).
perm :: Spe a [[a]]

-- | The species of k element subsets.
kSubset :: Int -> Spe a [a]

-- | The species of subsets. The definition given here is equivalent to
--   <tt>subset = map fst . (set .*. set)</tt>, but a bit faster.
subset :: Spe a [a]