MATH 145: HOMEWORK 3
ANDREW BERGET
This homework is due on Wednesday January 27. In these problem you will need to use the fact that
X
n
≥
0
ax
n
=
a
1

x
.
Read section 5.3 and solve problems 5.23(a) and 5.25+.
Problem A.
In this problem we consider the generating function for the Stirling numbers of the second
kind,
S
(
n,k
). Fix
k
≥
0. Deﬁne
G
k
(
x
) =
X
n
≥
0
S
(
n,k
)
x
n
(1) Consider the recurrence relation for
S
(
n,k
):
S
(
n,k
) =
S
(
n

1
,k

1) +
kS
(
n

1
,k
)
,
n,k
≥
1
.
Show this gives rise to a recurrnce relation for
G
k
(
x
):
G
k
(
x
) =
xG
k

1
(
x
) +
kxG
k
(
x
)
.
(2) Prove by induction on
k
that
G
k
(
x
) =
x
1

kx
x
1

(
k

1)
x
...
x
1

x
G
0
(
x
)
and determine what
G
0
(
x
) is in order to determine what
G
k
(
x
) is.
(3) + Explain why your answer is correct for
k
= 1. Write down a formula for
S
(
n,
2)?
Problem B.
+ In how many ways can we distribute 8 balls into 6 boxes if
(1) The balls and boxes are all distinct?
(2) The balls and boxes are all distinct, box #1 contains 2 balls, box #2 contains 3 balls, box #3
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 Winter '07
 Peche
 Math, Combinatorics, Recursion, Recurrence relation, Generating function

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