Due: Friday, May 21
Math 239
Assignment 2
1. Express the following coefficients as summations involving only ordinary binomial coef
ficients.
(a) (4 points) [
x
n
](1

2
x
)

m
(1

x
2
)
m
.
(b) (4 points) [
x
n
](1 + 2
x

3
x
2
)

4
.
(c) (4 points) [
x
n
](1 + 2
x
)

5
(1

x
3
)
m
.
2. Let
x
1
, . . . , x
k
be variables. A
monomial
in these variables is a product
x
m
1
1
· · ·
x
m
k
k
where
m
i
’s are nonnegative integers.
(For example monomials in
x
1
and
x
2
include
1,
x
5
2
and
x
2
1
x
2
.) The
degree
of a monomial is the sum of the exponents—in this case
m
1
+
· · ·
+
m
k
. Let
N
denote the set of nonnegative integers.
(a) (2 points) Define a weight function on
N
k
such that the number of elements in
N
k
with weight
m
is equal to the number of monomials in
x
1
, . . . , x
k
with degree exactly
m
.
(b) (3 points) Compute the generating series for
N
k
.
(c) (3 points) Use the binomial theorem to compute the number of monomials in
k
variable with degree exactly
n
.
3. Let
S
be the set of all subsets of
{
0
, . . . , n

1
}
and define the weight of a subset to be
the sum of its elements. Let
T
be the subset of
S
consisting of the subsets that do not
contain
n

1. Let Φ
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 M.PEI
 Math, Taylor Series, Binomial, Summation, Exponentiation

Click to edit the document details