Partitional/Cauchy product

The partitional product F ·G of two species F and G consists of paired F- and G- shapes, as with the Cartesian product, but with the labels partitioned between the two shapes instead of replicated (Figure 4.5). The divided box with rounded corners used in Figure 4.5 will often be used to schematically indicate a partitional product.

Definition 4.2.1. The partitional or Cauchy product of two speciesF and G is the functor defined on objects by

(F ·G) L= ]

LF,LG`L

F LF ×G LG

where U

denotes an indexed coproduct (i.e. disjoint union) of sets, and LF, LG ` L

indicates thatLF andLG constitute a partition ofL, (i.e.LF∪LG =LandLF∩LG=

∅); note that LF and LGmay be empty. In words, an (F·G)-shape with labels taken

fromL consists of some partition of L into two disjoint subsets, with anF-shape on one subset and a G-shape on the other.

7 4 5 6 2 1 0 3

On morphisms, (F ·G) σ is the function

(LF, LG, x, y)7→(σ LF, σ LG, F (σ|LF) x, G(σ|LG) y),

whereLF, LG `Land x∈F LF and y∈G LG. That is,σ acts independently on the

two subsets of L.

To compute the ogf of a product species F ·G, consider the product of ogfs

e F(x)Ge(x) = X n>0 fnxn ! X n>0 gnxn ! =X n>0 X 06k6n fkgn−k ! xn.

Note that the inner sum P

06k6nfkgn−k is indeed the number of (F ·G)-forms of size

n: such forms necessarily consist of an F-form of size k paired with a G-form of size

n−k. Hence

^

(F ·G)(x) = Fe(x)Ge(x).

The computation of the egf of a product species is similar.

F(x)G(x) = X n>0 fn xn n! ! X n>0 gn xn n! ! =X n>0 X 06k6n fk k! gn−k (n−k)! ! xn =X n>0 X 06k6n n k fkgn−k ! xn n!. Again, we verify that the inner sum P

06k6n n k

fkgn−k is the number of labelled

(F ·G)-shapes of size n: for each 0 ₆ k ₆ n, there are n_k ways to partition the n

labels into two subsets of size k and n−k, and then there are fk ways to make an

F-shape on the first subset, and gn−k ways to make a G-shape on the second. We

therefore have

(F ·G)(x) =F(x)G(x) as well.

The identity for partitional product should evidently be some species 1such that (1·G)L= ]

LF,LG`L

1LF ×G LG

!

'G L.

The only way for this isomorphism to hold naturally in L is if 1_∅={?} (yielding a summand of G L when _∅, L `L) and 1 LF =∅ for all other LF (cancelling all the

Figure 4.6: Permutation = fixpoints· derangement

Definition 4.2.2. The unit species, 1, is defined by

1 L=

(

{?} L=_∅

∅ otherwise.

Remark. Recall that one should not think of1as doing case analysis. Rather, a more

intuitive way to think of it is “there is a single 1-shape, and it has no labels”; that is, the unit species denotes a sort of “trivial” or “leaf” structure containing no labels. Intuitively, it corresponds to a Haskell type like

dataUnita =Unit

The generating functions for 1are given by

1(x) =e1(x) = 1.

Example. The following example is due to Joyal [1981]. Recall that S denotes the

species of permutations. Consider the species Der of derangements, that is, permu- tations which have no fixed points. It is not possible, in general, to directly express species using a “filter” operation, as in, “all F-shapes satisfying predicate P”. How- ever, it is possible to get a handle on Der in a more constructive manner by noting that every permutation can be canonically decomposed as a set of fixed points paired with a derangement on the rest of the elements (Figure 4.6). That is, algebraically,

S=E·Der. (4.2.1)

This does not directly give us an expression for Der, since there is no notion of mul- tiplicative inverse for species2. However, this is still a useful characterization of de- rangements. For example, since the mapping from species to egfs is a homomorphism

2_{Multiplicative inverses can in fact be defined for suitablevirtual species [Bergeron et al., 1998,}

with respect to product, (4.2.1) becomes 1 1−x =e

x_·

Der(x).

We can solve to obtain the egf Der(x) = e−x/(1−x), even though we cannot make direct combinatorial sense out of Der=S/E.

Proposition 4.2.3. (Spe,·,1) is symmetric monoidal.

Proof. We constructed 1so as to be an identity for partitional product. Associativity

and symmetry of partitional product are not hard to prove and are left as an exercise

for the reader. SDG

In fact, (Spe,·,1) is closed as well, but a discussion of the internal Hom functor corresponding to partitional product must be postponed to §4.5.5, after discussing species differentiation.