Combinatorics Through Guided Discovery, 2004a

C.4. The Product Principle for EGFs 159
· Problem 395. In Problem 385, why is the family of people actually making
phone calls (assuming nobody is calling outside the telephone network) at
any given time, with the added relationship of who is calling whome, a
species? Why is the family of sets of people who are not using their phones
a species (with no additional construction needed)?
The second essential feature of our examples of products of EGFs is that products
of EGFs seem to count structures on ordered pairs of two disjoint sets (or more
generally on k-tuples of mutually disjoint sets). For example, we can determine
a five coloring of a set S by partitioning it in all possible ways into two sets and
coloring the first set in the pair with our first two colors and our second pair with
the last three colors. Or we can partition our set in all possible ways into five parts
and color part i with our ith color. We don’t have to do the same thing to each
part of our partition; for example, we could define a derangement on one part and
an identity permutation on the other; this defines a permutation on the set we are
partitioning, and we have already noted that every permutation arises in this way.
Our combinatorial interpretation of EGFs will involve assuming that the coefficient
of xi
i!
counts the number of structures on a particular set of of size i in a species
of structures on subsets of a set X. Thus in order to give an interpretation of the
product of two EGFs we need to be able to think of ordered pairs of structures on
disjoint sets or k-tuples of structures on disjoint sets as structures themselves. Thus
given a structure on a set S and another structure on a disjoint set T, we define the
ordered pair of structures (which is a mathematical construction!) to be a structure
on the set S ∪ T. We call this a pair structure on S ∪ T. We can get many structures
on a set S ∪ T in this way, because S ∪ T can be divided into many other pairs of
disjoint sets. In particular, the set of pair structures whose first structure comes
from F and whose second element comes from G is denoted by F·G.
Problem 396. Show that if F and G are species of structures on subsets of
a set X, then the pair of structures of F·Gfor a species of structures
Given a species F of structures, the number of structures using any particular
set of size i is the same as the number of structures in the family using any other
set of size i. We can thus define the exponential generating function (EGF) for the
family as the power series ∑ ∞
i=1 a i xi
i! , where a i is the number of structures of F that
use one particular set of size i. In Problems 372, 373, 376, 377, 378, 380, 382, 383,
387, and 388, we were computaing EGFs for species of subsets of some set.
Problem 397. If F and G are species of subsets of X, how is the EGF for
F·Grelated to the EGFs for F and G? Prove you are right. (h)