Combinatorics Through Guided Discovery, 2004a

50 2. Applications of Induction and Recursion in Combinatorics and Graph Theory
(b) An alkane is called butane if it has four carbon atoms. Why do we say one
version of butane is shown in Figure 2.4.1?
4.
(a) Give a recurrence for the number of ways to divide 2n people into sets of two
for tennis games. (Don’t worry about who serves first.)
(b) Give a recurrence for the number of ways to divide 2n people into sets of two
for tennis games and to determine who serves first.
⇒
5. Give a recurrence for the number of ways to divide 4n people into sets of four
for games of bridge. (Don’t worry about how they sit around the bridge table or
who is the first dealer.)
6. Use induction to prove your result in Supplementary Problem 1.4.2 at the end
of Chapter 1.
n∏
7. Give an inductive definition of the product notation a i .
8. Using the fact that (ab) k = a k b k , use your inductive definition of product notation
in Problem 2.4.7 to prove that a i = a k i
( n
)
∏ k n∏
.
i=1
⇒ 9. How many labelled trees on n vertices have exactly four vertices of degree 1?
i=1
i=1
⇒∗
10. The degree sequence of a tree is a list of the degrees of the vertices in nonincreasing
order. For example the degree sequence of the first graph in Figure 2.3.3
is (4, 3, 2, 2, 1). For a graph with vertices labeled 1 through n, theordered degree
sequence of the graph is the sequence (d 1 , d 2 ,...,d n ) in which d i is the degree of
vertex i. For example, the ordered degree sqeuence of the first graph in Figure 2.3.1
is (1, 2, 3, 3, 1, 1, 2, 1).
(a) How many labelled trees are there on n vertices with ordered degree sequence
d 1 , d 2 ,...d n ? (This problem appears again in the next chapter since some ideas
in that chapter make it more straightforward.)
(b) How many labeled trees are there on n vertices with the degree sequence in
which the degree d appears i d times?