This preview shows pages 1–4. Sign up to view the full content.
MATH 239 Final Exam, Dec. 20, 2004
2
1.
(a)
[2 marks]
List all compositions of 2, 3 and 4 where each part is a positive integer not
equal to 2.
(b)
[2 marks]
Prove that the generating function for the number of compositions of a
positive integer
n
into
k
parts, such that each part is a positive integer which is not
equal to two, is
Φ(
x
) =
±
x

x
2
+
x
3
1

x
²
k
.
(c)
[2 marks]
Deteremine the generating function for the number,
a
n
, of compositions of
n
with no restriction on the number of parts in the composition. As in parts (a) and (b),
each part is a positive integer which is not equal to 2.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document MATH 239 Final Exam, Dec. 20, 2004
3
(d)
[3 marks]
Let
{
b
n
}
be the sequence whose generating function is deﬁned by
X
n
≥
0
b
n
x
n
=
1 +
x
4
1

x

2
x
3
.
Determine a recurrence relation, together with suﬃcient initial conditions, to uniquely
determine the sequence of
b
n
’s.
(e)
[3 marks]
The recurrence relation
c
n
= 5
c
n

1

6
c
n

2
for all integers
n
≥
2 with initial
conditions
c
0
=

3 and
c
1
=

4, deﬁnes the terms of a sequence
{
c
n
}
. Solve this
recurrence relation.
MATH 239 Final Exam, Dec. 20, 2004
4
2.
(a) For each of the following sets, write down a decomposition that uniquely creates the
elements of that set.
i.
[2 marks]
The set of
{
0
,
1
}
strings where all blocks have even length.
ii.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 07/28/2009 for the course MATH math239 taught by Professor ... during the Spring '06 term at Waterloo.
 Spring '06
 ...
 Math

Click to edit the document details