Combinatorics Through Guided Discovery, 2004a

158 C. Exponential Generating Functions
· Problem 393. Suppose that a i is the number of ways to paint a set of i poles
with red and white, and b j is the number of ways to paint a set of j poles
with blue, green, and yellow. In how many ways may we take a set N of n
poles, divide it up into two sets I and J (using i to stand for the size of I and
j to stand for the size of the set J, and allowing i and j to vary) and paint the
poles in I red and white and the poles in J blue, green, and yellow? (Give
your answer in terms of a i and b j . Don’t figure out formulas for a i and b j to
use in your answer; that will make it harder to get the point of the problem!)
How does this relate to Problem 392?
Problem 393 shows that the formula you got for the coefficient of xn
n!
in the
product of two EGFs is the formula we get by splitting a set N of poles into
two parts and painting the poles in the first part with red and white and the
poles in the second part with blue, green, and yellow. More generally, you could
interpret your result in Problem 392 to say that the coefficient of xn
n!
in the product
∑ ∞
i=0 a ∑
i xi ∞
i! j=0 b j x j
j!
of two EGFs is the sum, over all ways of splitting a set N of size
n into an ordered pair of disjoint sets I and J, of the product a |I| b |J| .
There seem to be two essential features that relate to the product of exponential
generating functions. First, we are considering structures that consist of a set and
some additional mathematical construction on or relationship among the elements
of that set. For example, our set might be a set of light poles and the additional
construction might be a coloring function defined on that set. Other examples of
additional mathematical constructions or relationships on a set could include a
permutation of that set; in particular an involution or a derangement, a partition
of that set, a graph on that set, a connected graph on that set, an arrangement of
the elements of that set around a circle, or an arrangement of the elements of that
set on the shelves of a bookcase. In fact a set with no additional construction or
arrangement on it is also an example of a structure. Its additional construction is the
empty set! When a structure consists of the set S plus the additional construction,
we say the structure uses S. What all the examples we have mentioned in our earlier
discussion of exponential generating functions have in common is that the number
of structures that use a given set is determined by the size of that set. We will call a
family F of structures a species of structures on subsets of a set X if structures are
defined on finite subsets of X and if the number of structures in the family using a
finite set S is finite and is determined by the size of S (that is, if there is a bijection
between subsets S and T of X, the number of structures in the family that use S
equals the number of structures in the family that use T). We say a structure is an
F -structure if it is a member of the family F .
Problem 394. · In Problem 383, why is the family of arrangements of set of
books on a single shelf (assuming they all fit) a species?