Combinatorics Through Guided Discovery, 2004a

166 C. Exponential Generating Functions
⇒ ·
Problem 421. Suppose that f (x) = ∑ ∞ x
n=0 c n
n n!
is the exponential generating
function for the number of simple connected graphs on n vertices and
g(x) = ∑ ∞
i=0 a i xi
i!
is the exponential generating function for the number of
simple graphs on i vertices. From this point onward, any use of the word
graph means simple graph.
(a) Is f (x) =e g(x) ,is f (x) =e g(x)−1 ,isg(x) =e f (x)−1 or is g(x) =e f (x) ? (h)
(b) One of a i and c n can be computed by recognizing that a simple graph
on a vertex set V is completely determined by its edge set and its edge
set is a subset of the set of two element subsets of V. Figure out which
it is and compute it. (h)
(c) Write g(x) in terms of the natural logarithm of f (x) or f (x) in terms
of the natural logarithm of g(x).
(d) Write (1 + y) as a power series in y. (h)
(e) Why is the coefficient of x0
0!
in g(x) equal to one? Write f (x) asapower
series in g(x) − 1.
(f) You can now use the previous parts of the problem to find a formula
for c n that involves summing over all partitions of the integer n. (It
isn’t the simplest formula in the world, and it isn’t the easiest formula
in the world to figure out, but it is nonetheless a formula with which
one could actually make computations!) Find such a formula. (h)
The point to the last problem is that we can use the exponential formula in
reverse to say that if g(x) is the generating function for the number of (nonempty)
connected structures of size n in a given family of combinatorial structures and
f (x) is the generating function for all the structures of size n in that family, then
not only is f (x) =e g(x) , but g(x) =( f (x)) as well. Further, if we happen to have
a formula for either the coefficients of f (x) or the coefficients of g(x), we can get a
formula for the coefficients of the other one!
C.6 Supplementary Problems
1. Use product principle for EGFs and the idea of coloring a set in two colors to
prove the formula e x · e x = e 2x .
2. Find the EGF for the number of ordered functions from a k-element set to an
n-element set.
3. Find the EGF for the number of ways to string n distinct beads onto a necklace.
4. Find the exponential generating function for the number of broken permutations
of a k-element set into n parts.
5. Find the EGF for the total number of broken permutations of a k-element set.