Math.Combinatorics.Species.Algebra

A data structure to reify combinatorial species.

Documentation

data SpeciesAlgT s where

Reified combinatorial species. Note that SpeciesAlgT has a phantom type parameter which also reflects the structure, so we can do case analysis on species at both the value and type level.

Of course, the non-uniform type parameter means that SpeciesAlgT cannot be an instance of the Species class; for that purpose the existential wrapper SpeciesAlg is provided.

Constructors

O :: SpeciesAlgT Z

I :: SpeciesAlgT (S Z)

X :: SpeciesAlgT X

:+: :: (ShowF (StructureF f), ShowF (StructureF g)) => SpeciesAlgT f -> SpeciesAlgT g -> SpeciesAlgT (f :+: g)

:*: :: (ShowF (StructureF f), ShowF (StructureF g)) => SpeciesAlgT f -> SpeciesAlgT g -> SpeciesAlgT (f :*: g)

:.: :: (ShowF (StructureF f), ShowF (StructureF g)) => SpeciesAlgT f -> SpeciesAlgT g -> SpeciesAlgT (f :.: g)

Der :: ShowF (StructureF f) => SpeciesAlgT f -> SpeciesAlgT (Der f)

E :: SpeciesAlgT E

C :: SpeciesAlgT C

OfSize :: SpeciesAlgT f -> (Integer -> Bool) -> SpeciesAlgT f

OfSizeExactly :: SpeciesAlgT f -> Integer -> SpeciesAlgT f

Instances

Show (SpeciesAlgT s)

data SpeciesAlg where

An existential wrapper to hide the phantom type parameter to SpeciesAlgT, so we can make it an instance of Species.

Constructors

SA :: ShowF (StructureF s) => SpeciesAlgT s -> SpeciesAlg

Instances

Show SpeciesAlg

C SpeciesAlg

C SpeciesAlg

C SpeciesAlg

Species SpeciesAlg

needsZT :: SpeciesAlgT s -> Bool

needsZT is a predicate which checks whether a species uses any of the operations which are not supported directly by ordinary generating functions (composition and differentiation), and hence need cycle index series.

needsZ :: SpeciesAlg -> Bool

A version of needsZT for SpeciesAlg.

reify :: SpeciesAlg -> SpeciesAlg

Reify a species expression into a tree. Of course, this is just the identity function with a usefully restricted type. For example:

 > reify octopus
 (C . C'_+)

reflectT :: Species s => SpeciesAlgT f -> s

Reflect a species back into any instance of the Species class.

reflect :: Species s => SpeciesAlg -> s

A version of reflectT for the existential wrapper SpeciesAlg.