Combinatorics Through Guided Discovery, 2004a

C.5. The Exponential Formula 163
and a nonzero number of one-cycles). What is f (x)? What is g(x)? How
are these to exponential generating functions related?
· Problem 412. Let f (x) be the EGF for the number of permutations of an
n-element set that have exactly one cycle. (This is the same as the EGF for
the number of ways to arrange n people around a round table.) Let g(x) be
the EGF for the total number of permutations of an n-element set. What is
f (x)? What is g(x)? How are f (x) and g(x) related?
There was one element that our last four problems had in common. In each
case our EGF f (x) involved the number of structures of a certain type (partitions,
telephone networks, tennis pairings, permutations) that used only one set of an
appropriate kind. (That is, we had a partition with one part, a telephone network
consisting either of one person or two people connected to each other, a tennis
pairing of one set of two people, or a permutation with one cycle.) Our EGF g(x)
was the number of structures of the same “type” (we put type in quotation marks
here because we don’t plan to define it formally) that could consist of any number
of sets of the appropriate kind. Notice that the order of these sets was irrelevant.
For example we don’t order the blocks of a partition and a product of disjoint cycles
is the same no matter what order we use to write down the product. Thus we were
relating the EGF for structures which were somehow “building blocks” to the EGF
for structures which were sets of building blocks. For a reason that you will see
later, it is common to call the building blocks connected structures. Notice that
our connected structures were all based on nonempty sets, so we had no connected
structures whose value was the empty set. Thus in each case, if f (x) = ∑ ∞
i=0 a i xi
i! ,
we would have a 0 =0. The relationship between the EGFs was always g(x) =e f (x) .
We now give a combinatorial explanation for this relationship.
· Problem 413. Suppose that F is a species of structures of a set X with no
structures on the empty set. Let f (x) be the EGF for F .
(a) In the power series
e f (x) =1+ f (x)+ f (x)2
2!
+ ···+ f (x)k
k!
+ ···=
∞∑
k=0
f (x) k
,
k!
what does Corollary C.4.3 tell us about the coefficient of xn
n!
(b) What does the coefficient of xn
n! in e f (x) count?
in
f (x)k
k!
?
In Problem 413 we proved the following theorem, which is called the exponential
formula.