Homework assignment Number 3
Math 145 // due on 01/26/2007
1. Problems Section 3.8: 6, 8, 9, 10, 13, 15.
2. A “walk” starts at the origin and makes up (resp. down) steps (1
,
1) (resp. (1
,

1)).
(a) How many walks are there from the origin to (
p, q
) for given integers
p >
0,
q
?
(b) Suppose now that the walk starts at
A
= (0
,
0) and finishes at
B
= (2
n,
0). We want
to determine the number of such walks that go below the
x
axis before 2
n
. The aim is
to show that this number is equal to the number of walks starting at
A
and ending at
C
= (2
n,

2).
i. Let
W
be a walk going below the
x
axis.
Suppose that
W
reaches the level

1
for the first time at the instant
n <
2
n
. Set
W
1
to be the trajectory which is the
reflection with respect to the axis
y
=

1 of
W
. Make a plot of two such trajectories
W
and
W
1
. What are the endpoints of
W
1
?
ii. Consider then the walk
W
which first follows the trajectory of
W
until time
n
and
then follows the same trajectory as
W
1
until time 2
n
. Show that
W
→
W
defines
a bijection between the set of walks from
A
to
B
going below the
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 Spring '07
 Peche
 Math, Combinatorics, Natural number, Mississippi, Bijection

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