Combinatorics Through Guided Discovery, 2004a

80 4. Generating Functions
• Problem 195. Find a formula for (1 + x) −n as a power series whose coefficients
involve binomial coefficients. What does this formula tell you about
how we should define ( −n )
k
when n is positive? (h)
Problem 196. If you define ( −n )
k
in the way you described in Problem 195,
you can write down a version of the binomial theorem for (x + y) n that is
valid for both nonnegative and negative values of n. Do so. This is called
the extended binomial theorem. Write down a special case with n negative,
like n = −3, to see an interesting surprise that suggests why we do not use
this formula later on.
Problem 197. Write down the generating function for the number of ways
to distribute identical pieces of candy to three children so that no child
gets more than 4 pieces. Write this generating function as a quotient of
polynomials. Using both the extended binomial theorem and the original
binomial theorem, find out in how many ways we can pass out exactly ten
pieces. (h)
• Problem 198. What is the generating function for the number of multisets
chosen from an n-element set so that each element appears at least j times
and less than m times? Write this generating function as a quotient of
polynomials, then as a product of a polynomial and a power series. (h)
⇒
Problem 199. Recall that a tree is determined by its edge set. Suppose you
have a tree on n vertices, say with vertex set [n]. We can use x i as the picture
of vertex i and x i x j as the picture of the edge x i x j . Then one possible picture
of the tree T is the product P(T) = ∏ {i,j}:i and j are adjacent x i x j .
(a) Explain why the picture of a tree is also ∏ n
i=1 x(i) .
i
(b) Write down the picture enumerators for trees on two, three, and four
vertices. Factor them as completely as possible.
(c) Explain why x 1 x 2 ···x n is a factor of the picture of a tree on n vertices.
(d) Write down the picture of a tree on five vertices with one vertex of
degree four, say vertex i. If a tree on five vertices has a vertex of degree
three, what are the possible degrees of the other vertices. What can
you say about the picture of a tree with a vertex of degree three? If
a tree on five vertices has no vertices of degree three or four, how