Due: Friday noon, May 27.
Each question carries 20 marks.
Math 239
Assignment 3
1. How many nonnegative integers smaller than 100,000 have the sum of their digits equal to
23?
2. Consider compositions of a positive integer
n
into 2 positive parts.
(a) Define an appropriate set
S
1
and write down a generating function
Φ
S
1
(
x
) to count the
number of compositions
n
=
c
+
d
with
c
<
d
. Express your answer as a rational function.
(b) Define another appropriate set
S
2
and write down another generating function
Φ
S
2
(
x
)
to count the number of compositions
n
=
e
+
f
with even
e
and arbitrary
f
. Express your
answer as a rational function.
(c) Suppose
Φ
S
1
(
x
)
=
∑
n
≥
1
a
n
x
n
and
Φ
S
2
(
x
)
=
∑
n
≥
1
b
n
x
n
. Use your answers in (a) and (b) to
show that
∀
n a
n
=
b
n
. Express
a
n
in terms of
n
.
(d) For each
n
, let
S
1
n
and
S
2
n
be the respective subsets of
S
1
and
S
2
containing the compo
sitions of
n
. Establish a bijection between
S
1
n
and
S
2
n
(which maps any (
c
,
d
)
∈
S
1
n
to some
(
e
,
f
)
∈
S
2
n
).
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 Spring '09
 M.PEI
 Integers, Natural number, Rational function, generating function ΦS

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