MATH 681
Problem Set #3
This problem set is due at the beginning of class on
October 1
.
1.
(10 points)
Prove the following combinatorial identities:
(a)
(5 points)
Recall that
S
(
n,k
) is equal to the number of ways to subdivide an
n
element set into
k
nonempty parts. Produce a combinatorial argument to show
that
S
(
n,k
) =
kS
(
n

1
,k
) +
S
(
n

1
,k

1).
(b)
(5 points)
Prove that for any
m < n
,
∑
m
k
=0
(
n
k,m

k,n

m
)
= 2
m
(
n
m
)
.
2.
(10 points)
We know that
p
k
(
n
) is the number of partitions of the number
n
into
exactly
k
nonzero parts, and it has a generating function given by
∑
∞
n
=0
p
k
(
n
)
x
n
=
x
k
(1

x
)(1

x
2
)(1

x
3
)(1

x
4
)
···
(1

x
k
)
.
(a)
(5 points)
Prove that
p
k
(
n
) =
p
k

1
(
n

1) +
p
k
(
n

k
) by using a direct combi
natorial method (e.g. bijection or alternative enumerations of the same set).
(b)
(5 points)
Prove that
p
k
(
n
) =
p
k

1
(
n

1)+
p
k
(
n

k
) by equating the generating
function
∑
∞
n
=0
p
k
(
n
)
x
n
to the sum
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 Fall '09
 WILDSTROM
 Math, Natural number, Recurrence relation, Fibonacci number

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