Math 184A
Second Exam (125 points)
26 February 2001
•
Please put your name and ID number on your exam. If you are not using a blue book,
put your name on every page.
•
The exam is CLOSED BOOK, but ONE SHEET of notes is allowed.
•
You must show your work to receive credit.
1. (45 pts.) (a) Compute the rank of the permutation 1
,
5
,
3
,
4
,
2 when the permutations
of
{
1
,
2
,
3
,
4
,
5
}
are listed in direct insertion order.
(b) What permutation immediately follows 1
,
5
,
3
,
4
,
2 in direct insertion order?
(c) What permutation immediately follows 1
,
5
,
3
,
4
,
2 in lex order?
2. (20 pts.) Find the unlabeled rooted plane tree with 7 leaves and rank 60.
b
1
=
b
2
= 1
,
b
3
= 2
,
b
4
= 5
,
b
5
= 14
,
b
6
= 42
,
b
7
= 132
.
3. (30 pts.) Let
C
n
(
k
) be the number of times position
k
is changed in the Gray code
for
n
long vectors of zeroes and ones given in the text. You may use the following fact
without proving it:
C
n
(
k
) =
(
1
if
k
= 1,
2
C
n

1
(
k

1)
if 1
< k
≤
n
,
0
if
k > n
.
(a) Tabulate values of
C
n
(
k
) for 1
≤
k
≤
n
≤
4.
(b) State and prove a simple formula for
C
n
(
k
) that does not involve a recursion.
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