Combinatorics Through Guided Discovery, 2004a

160 C. Exponential Generating Functions
Problem 398. Without giving the proof, how can you compute the EGF f (x)
for the number of structures using a set of size n in the species F 1 ·F 2 ·····F k
of structures on k-tuples of subsets of X from the EGFs f i (x) for F i for each
i from 1 to k? (Here we are using the natural extension of the idea of the
pair of structures to the idea of a k-tuple structure.)
Theorem C.4.1. If F 1 , F 2 ,...,F n are species of subsets of the set X and F i has EGF f i (x),
then the family of k-tuple structures F 1 ·F 2 ·····F n has EGF ∏ n
i=1 f i(x).
We call Theorem C.4.1 the product principle for exponential generating functions.
We give two corollaries; the proof of the second is not immediate though
not particular difficult.
Corollary C.4.2. If F is a species of structures on subsets of X and f (x0) is the EGF
for F , then f (x) k is the EGF for the k-tuple structure on k-tuples of F -structures using
disjoint subsets of X.
Our next corollary uses the idea of a k-set structure. Suppose we have a species
F of structures on nonempty subsets of X, that is, a species of structures which
assigns no structures to the empty set. Then we can define a new species F (k) of
structures, called “k-set structures,” using nonempty subsets of X. Given a fixed
positive integer k, ak-set structure on a subset Y of X consists of a k-element
set of nonempty disjoint subsets of X whose union is Y and an assignment of an
F -structure to each of the disjoint subsets. This is a species on the set of subsets of
X; the subset used by a k-set structure is the union of the sets of the structure. To
recapitulate, the set of k-set structures on a subset Y of X is the set of all possible
assignments of F -structures to k nonempty disjoint sets whose union is Y. (You
can also think of the k-set structures as a family of structures defined on blocks of
partitions of subsets of X into k blocks.)
Corollary C.4.3. If F is a species of structures on nonempty subsets of X and f (x) is the
f (x)
EGF for F , then for each positive integer k,
k
is the EGF for the family F (k) of k-set
k!
structures on subsets of X
Problem 399. Prove Corollary C.4.3. (h)
· Problem 400. Use the product principle for EGFs to explain the results of
Problems 390 and Problem 391.
Problem 401. · Use the general product principle for EGFs or one of its corollaries
to explain the relationship between the EGF for painting streetlight
poles in only one color and the EGF for painting streetlight poles in 4 colors