Combinatorics Through Guided Discovery, 2004a

191
391. Notice that any permutation is a product of a derangement of the elements
not fixed by the permutation times a permutation whose cycle decomposition
consists of one-cycles.
392. A binomial coefficient is likely to appear in your answer.
397. If f (x) = ∑ ∞
i=0 a i xi
i!
and g(x) = ∑ ∞
j=0 b j x j
j!
, what is the coefficient of
xn
n!
in
f (x)g(x)? don’t be surprised if your answer has a binomial coefficient in it.
In fact, the binomial coefficient should help you finish the problem.
399. Since the sets of a k-set structure are nonempty and disjoint, the k-element
set of sets can be arranged as a k-tuple in k! ways.
403. The alternate definition of a funciton in Section 3.1.2 can be restated to say
that a function from a k-element set K to an n-element set N can be thought of
as an n-tuple of sets, perhaps with some empty, whose union is K. In order to
think of the function as an n-tuple, we number the elements of N as number
1 through number n. Then the ith set in the n-tuple is the set of elements
mapped to the ith element of N in our numbering?
404. Don’t be surprised if you see a hyperbolic sine or hyperbolic cosine in your
answer. If you aren’t familiar with these functions, look them up in a calculus
book.
407. The EGF for ∑ n
i=1 (n k )k is ∑ ∞ ∑ n n!
n=1 i=1 k!(n−k)! k xn
n!
. You can cancel out the n!
terms and the k terms. Now try to see if what is left can be regarded as the
product of two EGFs.
421.a. To apply the exponential formula, we must take the exponential function
of an EGF whose constant term is zero, or in other words, for a species of
structures that has no structures that use the empty set.
421.b. Once you know the vertex set of a graph, all you have to do to specify the
graph is to specify its set of edges.
421.d. What is the calculus definition of (1 + y)?
421.f. Look for a formula that involves summing over all partitions of the integer n.