MATH 239 Assignment 3
This assignment is due at noon on Friday, June 8, 2007, in the drop boxes opposite the Tutorial
Centre, MC 4067.
1. (a) Determine a generating function for the compositions of
n
in which all parts are odd.
As usual, the weight of a composition is the sum of its parts.
(b) Determine a generating function for the of compositions of
n
in which at least one part
is even.
(c) Let the number of compositions of
n
with at least one even part be
b
n
. Determine a
recurrence relation for the
b
n
’s, and suﬃcient initial conditions to uniquely deﬁne the
terms of the sequence
{
b
n
}
. Evaluate
b
10
.
2. (a) Let
A
=
{
10
,
101
}
and let
B
=
{
001
,
100
,
1001
}
. For each of the sets
AB
and
BA
, de
termine whether or not the elements are uniquely created. Find the generating function
for
AB
and
BA
with respect to length.
(b) Let
A
=
{
00
,
101
,
11
}
and
B
=
{
00
,
001
,
10
,
110
}
. Prove that the elements of
A
*
are
uniquely created, but the strings of
B
*
are not. Find the generating function for
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 01/17/2011 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.
 Spring '09
 M.PEI
 Math

Click to edit the document details