Mathematics for Computer Science
Infinite Geometric Sum
MIT 6.042J/18.062J
S(x) ::= 1+ x + x2 + + xn + xS(x) = x + x2 + + xn + Generating Functions
Albert R Meyer,
lec 11F.1
April 23, 2010
Albert R Meyer,
April 23, 2010
lec 11F.2
Ordinary Generating Functions
Infinite Geometric Sum
S(x) ::= 1+ x + x2 + + xn + x + x2 + + x n + xS(x) =
The ordinary generating function
for the infinite sequence
g0, g1, g2, , gn, S(x)xS(x) = 1
is the power series:
G(x) = g0 + g1x + g2x2 + + gn xn + Albert R Meyer,
lec 11F.3
April 23, 2010
April 23, 2010
lec 11F.5
In how many ways can we fill a bag
with n fruits given the following
constraints?
“corresponds to”
Albert R Meyer,
April 23, 2010
Bags of fruit
Infinite Geometric Sum
=
Albert R Meyer,
• At most 2 oranges.
1
• Any number of apples.
1x
• Any number of bananas that only
come in bunches of 3.
lec 11F.6
Albert R Meyer,
April 23, 2010
lec 11F.8
1
At most 2 oranges
Bags with n = 4 fruits
# ways to pick
k oranges
• 0 oranges, 1 apple, 3 bananas
• 0 oranges, 4 apples, 0 bananas
• 1 orange, 0 apples, 3 bananas
• 1 orange, 3 apples, 0 bananas
• 2 oranges, 2 apples, 0 bananas
Number of 4-fruit bags: 5
Albert R Meyer,
April 23, 2010
lec 11F.9
Albert R Meyer,
April 23, 2010
lec 11F.10
Substituting xk for x
Any number of apples
There is only 1 way to pick a bag
of k apples: ak = 1
Albert R Meyer,
April 23, 2010
lec 11F.11
Albert R Meyer,
April 23, 2010
lec 11F.12
Convolution Rule
Bananas in bunches of 3
We can use
the individual
generating
functions to
solve original
fruit problem!
Albert R Meyer,
April 23, 2010
lec 11F.13
Albert R Meyer,
April 23, 2010
lec 11F.14
2
Convolution Rule
Convolution Rule
Ways to pick 12 apples & bananas:
# ways
• 0 apples and 12 bananas
1
• 1 apple and 11 bananas
0
• 12 apples and 0 bananas
Ways to pick 12 apples & bananas:
aj
= # ways to pick j apples
bk
= # ways to pick k bananas
ajb12-j = # ways to pick j apples
and rest bananas
1
Total=5
Albert R Meyer,
April 23, 2010
lec 11F.15
Convolution Rule
Albert R Meyer,
April 23, 2010
lec 11F.16
Convolution Rule
# ways to pick 12 apples & bananas
The coefficient of x12 in the product
A(x)·B(x):
= a 0b 12 + a 1b 11 + … + a 11b 1 + a 12b 0
But this is the coefficient
of x12 in A(x)·B(x)
Albert R Meyer,
April 23, 2010
lec 11F.17
Convolution Rule
Albert R Meyer,
April 23, 2010
lec 11F.18
Convolution Rule
# ways to pick 12 apples & bananas
The gen func for choosing from
a union of disjoint sets is the
product of the gen funcs for
choosing from each set.
is the coefficient of x12 in
A(x)·B(x)
the generating function for
picking apples & bananas
Albert R Meyer,
April 23, 2010
lec 11F.19
Albert R Meyer,
April 23, 2010
lec 11F.20
3
Bags of Fruit
A Familiar Generating Function?
Gen func for the bags of fruit:
so # of our bags with n fruits is
the coefficient of xn in
1 / (1 x )
2
=
We can easily relate 1/(1-x)2
to something we already know
how to count!
1
(1 x )
2
Albert R Meyer,
April 23, 2010
lec 11F.21
A Familiar Generating Function?
The gen func for selecting n
donuts of a given flavor:
Albert R Meyer,
April 23, 2010
Albert R Meyer,
April 23, 2010
lec 11F.22
A Familiar Generating Function?
The gen func for selecting n
donuts using both flavors:
lec 11F.24
A Familiar Generating Function?
The gen func for selecting n
donuts among k flavors:
Albert R Meyer,
April 23, 2010
lec 11F.25
The Donut Number!
Using k different flavors, how
many ways are there to form a
bag of n donuts?
(You already know the answer
to this one.)
Albert R Meyer,
April 23, 2010
lec 11F.26
Albert R Meyer,
April 23, 2010
lec 11F.27
4
The Donut Number!
The Donut Number!
Using k different flavors, how
many ways are there to form a
bag of n donuts?
so coeff of xn in
n + k 1
n Albert R Meyer,
n + k 1
n lec 11F.28
April 23, 2010
Conclusion: Bags of Fruit
(1 x )
2
fn = n + 2 1 = n + 1
n lec 11F.30
April 23, 2010
Partial Fraction Expansions
H(x) ::=
Albert R Meyer,
April 23, 2010
lec 11F.34
Partial Fractions for H(x)
x
2x2 3x + 1
lec 11F.29
If a generating function
polynomials
H(x) is a quotient
rational of
function
there is a simple way to find
the nth coefficient hn
1
Albert R Meyer,
April 23, 2010
Finding coefficients
In how many ways can we fill a
bag with n of our fruits?
F(x) =
Albert R Meyer,
Factor
denominator
1 1 H(x) = A 1 + A2 1 2x 1 x h n = A 1 2n + A 2 1
Express as
sum
TO DO: find A1 and A2.
Albert R Meyer,
April 23, 2010
lec 11F.35
Albert R Meyer,
April 23, 2010
lec 11F.36
5
Solve for A1 and A2
Solve for A1 and A2
Substitute in values for x.
Multiply both sides by denom of LHS.
Albert R Meyer,
lec 11F.37
April 23, 2010
Finding the coefficients
1
=
1 2x
April 23, 2010
lec 11F.38
In General…
1
The partial fraction expansion of
P(x)/Q(x) contains terms of the form
1x
the partial fraction expansion
h n = 2n 1
Albert R Meyer,
Albert R Meyer,
…+
A
(1 x )
k
+…
We know the nth
coeff of this!
n + k - 1
n
A n where 1/
is a root of Q(x).
lec 11F.39
April 23, 2010
Partial Fractions Caveat #1
Albert R Meyer,
April 23, 2010
lec 11F.40
Partial Fractions Caveat #2
For roots with multiplicity k>1 in
factored denominator of gen func
(1 x ) If deg(N) > deg(D)…
use polynomial long division to find
Q(x) and R(x) such that
k
need k partial fractions:
A1
+
A2
(1 x ) (1 x )
1
2
Albert R Meyer,
++
Ak
(1 x )
April 23, 2010
k
+
lec 11F.41
and deg(R) < deg(D).
Albert R Meyer,
April 23, 2010
lec 11F.42
6
Team Problems
Problems
1&2
Albert R Meyer,
April 23, 2010
lec 11F.44
7
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6.042J / 18.062J Mathematics for Computer Science
Spring 2010
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