Combinatorics Through Guided Discovery, 2004a

154 C. Exponential Generating Functions
◦ Problem 379. What is the EGF for the sequence 0, 1, 2, 3,...? You may think
of this as the EFG for the number of ways to select one element from an n
element set. What is the EGF for the number of ways to select two elements
from an n-element set?
· Problem 380. What is the EGF for the sequence 1, 1,...,1,...? Notice that
we may think of this as the EGF for the number of identity permutations
on an n-element set, which is the same as the number of permutations of
n elements that are products of 1-cycles, or as the EGF for the number of
ways to select an n-element set (or, if you prefer, an empty set) from an n-
element set. As you may have guessed, there are many other combinatorial
interpretations we could give to this EGF.
◦ Problem 381. What is the EGF for the number of ways to select n distinct
elements from a one-element set? What is the EGF for the number of ways
to select a positive number n of elements from a one element set? Hint:
When you get the answer you will either say “of course,” or “this is a silly
problem.” (h)
· Problem 382. What is the EGF for the number of partitions of a k-element
set into exactly one block? (Hint: is there a partition of the empty set into
exactly one block?)
· Problem 383. What is the EGF for the number of ways to arrange k books
on one shelf (assuming they all fit)? What is the EGF for the number of
ways to arrange k books on a fixed number n of shelves, assuming that all
the books can fit on any one shelf? (Remember Problem 122.)
C.3 Applications to recurrences.
We saw that ordinary generating functions often play a role in solving recurrence
relations. We found them most useful in the constant coefficient case. Exponential
generating functions are useful in solving recurrence relations where the coefficients
involve simple functions of n, because the n! in the denominator can cancel
out factors of n in the numerator.