{-# LANGUAGE NoImplicitPrelude , GeneralizedNewtypeDeriving #-} ----------------------------------------------------------------------------- -- | -- Module : Math.Combinatorics.Species.Types -- Copyright : (c) Brent Yorgey 2010 -- License : BSD-style (see LICENSE) -- Maintainer : byorgey@cis.upenn.edu -- Stability : experimental -- -- Some common types used by the species library, along with some -- utility functions. -- ----------------------------------------------------------------------------- module Math.Combinatorics.Species.Types ( -- * Miscellaneous CycleType -- * Series types , EGF(..) , egfFromCoeffs , liftEGF , liftEGF2 , GF(..) , gfFromCoeffs , liftGF , liftGF2 , CycleIndex(..) , ciFromMonomials , liftCI , liftCI2 -- * Series utility functions , filterCoeffs , selectIndex ) where import NumericPrelude import PreludeBase import Data.List (genericReplicate) import qualified MathObj.PowerSeries as PS import qualified MathObj.MultiVarPolynomial as MVP import qualified MathObj.Monomial as Monomial import qualified Algebra.Additive as Additive import qualified Algebra.Ring as Ring import qualified Algebra.Differential as Differential import qualified Algebra.ZeroTestable as ZeroTestable import qualified Algebra.Field as Field -- | A representation of the cycle type of a permutation. If @c :: -- CycleType@ and @(k,n) ``elem`` c@, then the permutation has @n@ -- cycles of size @k@. type CycleType = [(Integer, Integer)] ------------------------------------------------------------ -- Series types ------------------------------------------ ------------------------------------------------------------ -- | Exponential generating functions, for counting labelled species. newtype EGF = EGF { unEGF :: PS.T Rational } deriving (Additive.C, Differential.C, Ring.C, Show) egfFromCoeffs :: [Rational] -> EGF egfFromCoeffs = EGF . PS.fromCoeffs liftEGF :: (PS.T Rational -> PS.T Rational) -> EGF -> EGF liftEGF f (EGF x) = EGF (f x) liftEGF2 :: (PS.T Rational -> PS.T Rational -> PS.T Rational) -> EGF -> EGF -> EGF liftEGF2 f (EGF x) (EGF y) = EGF (f x y) -- | Ordinary generating functions, for counting unlabelled species. newtype GF = GF (PS.T Integer) deriving (Additive.C, Ring.C, Show) gfFromCoeffs :: [Integer] -> GF gfFromCoeffs = GF . PS.fromCoeffs liftGF :: (PS.T Integer -> PS.T Integer) -> GF -> GF liftGF f (GF x) = GF (f x) liftGF2 :: (PS.T Integer -> PS.T Integer -> PS.T Integer) -> GF -> GF -> GF liftGF2 f (GF x) (GF y) = GF (f x y) -- | Cycle index series. newtype CycleIndex = CI (MVP.T Rational) deriving (Additive.C, Ring.C, Differential.C, Show) ciFromMonomials :: [Monomial.T Rational] -> CycleIndex ciFromMonomials = CI . MVP.Cons liftCI :: (MVP.T Rational -> MVP.T Rational) -> CycleIndex -> CycleIndex liftCI f (CI x) = CI (f x) liftCI2 :: (MVP.T Rational -> MVP.T Rational -> MVP.T Rational) -> CycleIndex -> CycleIndex -> CycleIndex liftCI2 f (CI x) (CI y) = CI (f x y) ------------------------------------------------------------ -- Some series utility functions ------------------------- ------------------------------------------------------------ -- | Filter the coefficients of a series according to a predicate. filterCoeffs :: (Additive.C a) => (Integer -> Bool) -> [a] -> [a] filterCoeffs p = zipWith (filterCoeff p) [0..] where filterCoeff p n x | p n = x | otherwise = Additive.zero -- | Set every coefficient of a series to 0 except the selected -- index. Truncate any trailing zeroes. selectIndex :: (Ring.C a, Eq a) => Integer -> [a] -> [a] selectIndex n xs = xs' where mx = safeIndex n xs safeIndex _ [] = Nothing safeIndex 0 (x:_) = Just x safeIndex n (_:xs) = safeIndex (n-1) xs xs' = case mx of Just 0 -> [] Just x -> genericReplicate n 0 ++ [x] _ -> []