Combinatorics Through Guided Discovery, 2004a

C.2. Exponential Generating Functions 153
◦ Problem 373. Find the EGF (exponential generating function) for the number
of ways to paint the n streetlight poles that run along the north side of
Main Street in Anytown, USA using four colors.
Problem 374. For what sequence is ex −e −x
2
= x the EGF (exponential
generating function)?
· Problem 375. For what sequence is ( 1
1−x
) the EGF? ((y) stands for the
natural logarithm of y. People often write (y) instead.) Hint: Think of
the definition of the logarithm as an integral, and don’t worry at this stage
whether or not the usual laws of calculus apply, just use them as if they do!
We will then define (1 − x) to be the power series you get. a
and
a It is possible to define the derivatives and integrals of power series by the formulas
d
∞∑ ∞∑
b i x i = ib i x i−1
dx
∫ x
0
i=0
i=0
i=1
∞∑ ∞∑
b i x i b i
=
i +1 xi+1
rather than by using the limit definitions from calculus. It is then possible to prove that the
sum rule, product rule, etc. apply. (There is a little technicality involving the meaning of
composition for power series that turns into a technicality involving the chain rule, but it
needn’t concern us at this time.)
i=0
· Problem 376. What is the EGF for the number of permutations of an n-
element set?
⇒ ·
Problem 377. What is the EGF for the number of ways to arrange n people
around a round table? Try to find a recognizable function represented by
the EGF. Notice that we may think of this as the EGF for the number of
permutations on n elements that are cycles. (h)
⇒ ·
Problem 378. What is the EGF ∑ ∞ x
n=0 p 2n
2n (2n)! for the number of ways p 2n to
pair up 2n people to play a total of n tennis matches (as in Problems 12 and
44)? (h)