Combinatorics Through Guided Discovery, 2004a

180 D. Hints to Selected Problems
though of as a function from the set [n − 2] to the set of vertices that do not
have degree 1. What is special about this function?
145. When you add the number of functions mapping onto J over all possible
subsets J of N, what is the set of functions whose size you are computing?
148. What if the j i ’s don’t add to k?
Additional Hint: Think about listing the elements of the k-element set and
labeling the first j 1 elements with label number 1.
149. The sum principle will help here.
150. How are the relevant j i ’s in the multinomial coefficients you use here different
from the j i ’s in the previous problem.
151. Think about how binomial coefficients relate to expanding a power of a binomial
and note that the binomial coefficient ( n )
k
and the multinomial coefficient
) are the same.
( n
k,n−k
152.a. We have related Stirling numbers to powers n k . How are binomial coefficients
related to falling factorial powers?
152.b. In the equation ∑ n
j=0 n j S(k, j) =n k , we might try substituting x for n. However
we don’t know what ∑ x
j=0
means when x is a variable. Is there anything
other than n that makes a suitable upper limit for the sum? (Think about
what you know about S(k, j).
153. For the last question, you might try taking advantage of the fact that x =
x +1− 1.
154. What does induction have to do with Equation (3.1)?
Additional Hint: What could you assume inductively about x k−1 if you were
trying to prove x k = ∑ k
n=0 s(k, n)x n ?
156.a. There is a solution for this problem similar to the solution to Problem 154.
156.b. Is the recurrence you got familiar?
156.d. Show that (−x) k =(−1) k x k and (−x) k =(−1) k x k .
Additional Hint: The first hint lets you write an equation for (−1) k x k as a rising
factorial of something else and then use what you know about expressing
rising factorials in terms of falling factorials, after which you have to convert
back to factorial powers of x.