The spe package — species lite

A simple library for combinatorial species with no dependencies but base. For a quick taste, look at the examples below. For further information, see the reference documentation on Hackage, and the full (but short!) source code: Math/Spe.hs.

Examples

Octopodes

An octopus is a cycle of nonempty lists:

oct = cyc `o` nonEmpty list

Connected lists

A connected list is a nonempty list that begins with its smallest element. E.g, [3,5,9,7] is connected but [2,4,1] is not. Using the ordinal product we can define the L-species of connected lists by

clist = x <*. list

in which x is the singleton species. Let us print a few of these connected lists to get a feeling for how this works:

> mapM_ print $ clist [2,5,6,9]
(2,[9,6,5])
(2,[9,5,6])
(2,[6,9,5])
(2,[6,5,9])
(2,[5,9,6])
(2,[5,6,9])

If we prefer to see these simply as lists rather than pairs, then we would define

clist = map (uncurry (:)) . (x <*. list)

Indeed, with this definition we have

> mapM_ print $ clist [2,5,6,9]
[2,9,6,5]
[2,9,5,6]
[2,6,9,5]
[2,6,5,9]
[2,5,9,6]
[2,5,6,9]

You might wonder why these lists are called "connected". It has to do with the species list and set `o` listc being isomorphic. In general, if f and g are species and f = set `o` g then g can be seen as (connected) components of f, and those components may be called connected f-structures.

Binary trees

Here's an example of a recursively defined species.

data BTree a = Empty | BNode a (BTree a) (BTree a) deriving (Show, Eq)

bTree :: Spe a (BTree a)
bTree [] = [ Empty ]
bTree xs = [ BNode v l r
           | (v,(l,r)) <- x .*. (bTree .*. bTree) $ xs
           ]

We could similary use the ordinal product to define increasing binary trees, also called min-heaps:

heap :: Spe a (BTree a)
heap [] = [ Empty ]
heap xs = [ BNode v l r
          | (v,(l,r)) <- x <*. (heap .*. heap) $ xs
          ]

Derivatives

The following expression evaluates to true and illustrates that the derivative of the species of cycles is isomorphic to the species of lists (linear orders):

(map catMaybes $ dx cyc [1..5]) == list [1..5]

Pointing

A pointed F-structure is an F-structure together with a distinguished element (the element pointed at):

> mapM_ print $ pointed set [1..3]
([1,2,3],1)
([1,2,3],2)
([1,2,3],3)

One may note that pointed f is isomorphic x .*. dx f. For example,

> mapM_ print $ (x .*. dx set) [1..3]
(3,[Nothing,Just 1,Just 2])
(2,[Nothing,Just 1,Just 3])
(1,[Nothing,Just 2,Just 3])

Cartesian product

The Cartesian product (><) of two species F and G on a set U is obtained by superimposing both an F-structure and a G-structure on U. For instance, pointing can be expressed using the Cartesian (and the ordinary) species product:

pointed f = f >< (x .*. set)

Functorial composition

The Spe package doesn't provide an operator for the functorial composition of species. This is because, for our definition of species, it coincides with the usual composition of functions. For instance, the species of simple undirected graphs can be defined like this:

graph = subset . kSubset 2

Ballot matrices

A ballot (or ordered set partition) is a list of blocks, where a block is simply a nonempty set. We may give ballots this type:

type Bal a = [[a]]

The ballot species can be defined by list `o` nonEmpty set. The type of this expression, Spe a ([[a]],[[a]]), is, however, a bit more complicated than we intended. Looking at the definition of partitional composition in Math/Spe.hs we realize that mapM (nonEmpty set) bs == bs for any set partition bs. Thus the second component is redundant, and a better definition of the species of ballots would be

bal :: Spe a (Bal a)
bal = map fst . (list `o` nonEmpty set)

The species of ballots is already defined in Math/Spe.hs. The definition given there is a bit different for reasons of efficiency.

In a recent preprint, Stuart Hannah and I study upper-triangular matrices of ballots without empty rows. Those can be defined as follows:

type Row a = [Bal a]
type BalMat a = [Row a]

rowOfLength :: Int -> Spe a (Row a)
rowOfLength i = bal .^ i

balMatOfDim :: Int -> Spe a (BalMat a)
balMatOfDim k = prod [ nonEmpty (rowOfLength i) | i <- [1..k] ]

balMat :: Spe a (BalMat a)
balMat xs = assemble [ balMatOfDim k | k <- [0..length xs] ] xs

Further, we define the sign of a ballot matrix by

sign :: BalMat a -> Int
sign m = (-1)^(dim m + blk m)
  where
    dim = length                   -- Matrix dimension
    blk = length . concat . concat -- Total number of blocks

Let us count ballot matrices with respect to this sign:

> [ sum . map sign $ balMat [1..n] | n<-[0..6] ]
[1,1,3,19,207,3451,81663]

Looking up this sequence in OEIS, here using the command line utility sloane, we find:

$ sloane 1,1,3,19,207,3451,81663

S A079144 1,1,3,19,207,3451,81663,2602699,107477247,5581680571,356046745023,
N A079144 Number of labeled interval orders on n elements: (2+2)-free posets.

In the aforementioned preprint we prove this surprising result using a sign reversing involution.

Notes

Transport of structure

Assuming that h . h' = h' . h = id, we can define the transport of structure as follows.

transport :: Functor f => (a -> b) -> (b -> a) -> Spe a (f a) -> Spe b (f b)
transport h h' spe = map (fmap h) . spe . map h'

For this to work consistently we would, however, have to define some additional newtypes and functor instances. For instance, in bal :: Spe a [[a]] we would like fmap to map two levels deep instead of one level deep in [[a]].

The species package by Brent Yorgey

This species module was created for use in my own research. It is sufficient for my needs. If you want something more substantial, then you will most likely be happier with the excellent species package by Brent Yorgey.