MATH 895-4 Fall 2010
Course Schedule
L ECTURE 2
f ac ulty of sc ience
depar tment of mathem atics
Generating functions
Generating functions
Week
Date
Sections
from FS2009
Part/ References
Contents
1
Sept 7 I.1, I.2, I.3
1
Combinatorial
Structures
FS: Part A.1, A.2
Enumeration
Comtet74
3
21
II.1, II.2, II.3
Handout #1 .
1.1 Generating functions
(self study)
4
28
II.4, II.5, II.6
2
14
Topic/Sections
Notes/Speaker
Symbolic methods
I.4, I.5, I.6
Unlabelled structures
Labelled structures II
2 Manipulating formal
power series
Combinatorial
Combinatorial
5
5
III.1, III.2
2.1Oct Definitions
. parameters
. . . . . . . . . .Parameters
. . . . . . . .
FS A.III
and
. . . . . . .Multivariable
. . . . .GFs. . .
6 2.212 Sum
IV.1,
IV.2 Product
(self-study)
2.3 Multiplicative inverse A(x)−1 . . . . . . . .
7
19
IV.3, IV.4
Complex Analysis
2.4 Composition Analytic
. . . Methods
. . . . . . . . . . . . . . .
FS: Part B: IV, V, VI
8
Appendix
2.526 Differentiation
. . .B4. . . . . .Singularity
. . . .Analysis
. . . .
IV.5 V.1
Stanley 99: Ch. 6
9 2.6NovTwo
2
special series:
exp, log . .Asymptotic
. . . . methods
. . . .
Handout #1
9
VI.1
12
A.3/ C
1
1
. . . .Labelled
. . . structures
. . . .I. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Asst
. .#1. Due
. .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. Asst
. .#2. Due
. .
(self-study)
3 10The basic toolbox for coefficient extraction
Introduction to Prob.
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2
2
3
3
3
4
4
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.
Sophie
4
Mariolys
18
IX.1
Limit Laws
andgenerating
Comb
Marni
4 A sampling
of counting sequences
and
functions
11
4.120 Advanced
Ideas
. .Structures
. . . . . .Discrete
. . . Limit
. . Laws
. . . . Sophie
. . . . . . . . . . . . . . . . . . . . . . . . . .
Random
IX.2
1
12
23
IX.3
and Limit Laws
FS: Part C
(rotating
presentations)
Enumeration
25
IX.4
Combinatorial
instances of discrete
Mariolys
Continuous Limit Laws
Marni
6
6
Quasi-Powers
question
about aandcombinatorial
Sophie
Gaussian limit laws
Perhaps
most fundamental
class, is “how many?”. The enumeration
13
30 theIX.5
problem can be extremely useful for solving other problems, and our approach will use generating
14
Dec 10since they are so extremely
Presentations powerful. We will
Asstonly
#3 Duejust scratch the surface of their potential.
functions
Our initial approach will build up a small toolbox of combinatorial constructions
• union
• cartesian product
• sequence
• set and multiset
• cycle
Dr. Marni
MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
• pointing and substitution
which translate to simple (and not so simple) operations on the generating functions.
1.1
Generating functions
We now jump head first into generating functions. Wilf ’s book generatingfunctionology is a great first
text if you want more details and examples.
Definition. The ordinary generating function (OGF) of a sequence (An ) is the formal power series
A(z) =
∞
X
An z n .
n=0
We extend this to say that the ogf of a class A is the generating function of its counting sequence
An = cardinality(An ). Equivalently, we can write the OGF as
X
A(z) =
z |α| .
α∈A
In this context we say that the variable z marks the size of the underlying objects.
M ARNI M ISHNA , S PRING 2011; K AREN Y EATS, S PRING 2013
MATH 343: A PPLIED D ISCRETE M ATHEMATICS
PAGE 1/6
MATH 895-4 Fall 2010
Course Schedule
L ECTURE 2
f ac ulty of sc ience
depar tment of mathem atics
Week
Date
1
Sections
Part/ References
Topic/Sections
Generating functions
Notes/Speaker
from FS2009
Exercise. Prove
that these two forms for A(z) are equivalent.
Sept 7
I.1, I.2, I.3
2
14
I.4, I.5, I.6
Symbolic methods
3
21
II.1, II.2, II.3
4
28
II.4, II.5, II.6
5
Oct 5
III.1, III.2
Combinatorial
For example, consider
the class of graphs H given here
Structures
FS: Part A.1, A.2
Comtet74
Handout #1
(self study)
Unlabelled structures
Labelled structures I
Labelled structures II
Combinatorial
parameters
Combinatorial
Parameters
Asst #1 Due
If we define size to be FS
the
A.IIInumber of vertices, then the generating function is simply
6
12
7
19
8
26
9
Nov 2
IV.1, IV.2
Multivariable GFs
(self-study)
4
4
H(z)
= z + z +Complex
z 3 + zAnalysis
+ z 1 + z 4 + z 3 = z + z 2 + 2z 3 + 3z 4 .
Analytic Methods
IV.3, IV.4
2
FS: Part B: IV, V, VI
Singularity Analysis
Appendix
B4 size to be the number of edges, then the ogf is
On the otherIV.5hand,
if we
define
V.1
10
9
VI.1
Stanley 99: Ch. 6
Handout #1
H(z)(self-study)
= z4 + z1
3
Asymptotic methods
6
0
Asst #2 Due
3
+ z + z + z + z + Sophie
z 2 = 1 + z + z 2 + 2z 3 + z 4 + z 6 .
12
A.3/ C
Introduction to Prob.
Mariolys
In both
these
cases we are treating the
generating function
as some sort of projection in which we
18 eachIX.1
Limit
Laws and
Comb
Marni
replace
vertex
(edge)
by
a
single
atomic
“z”
and
forget
all
detailed information about the object
11
Random
Structures together
and then
monomials
toLaws
get the
ogf.
20 just
IX.2gather the
Discrete Limit
Sophie
Limit Laws
Thus our examplesand
become
FS: Part C
Combinatorial
23
IX.3
25
IX.4
13
30
IX.5
14
Dec 10
12
(rotating
presentations)
instances of discrete
Mariolys
∞
X
Continuous Limit Laws Marni
Binary words:
W (z) =
2n z n =
Quasi-Powers and
Gaussian limit laws
Permutations:
Presentations
n=0
∞
X
Sophie
1
1 − 2z
n
P (z) =Asst #3 n!z
Due
n=0
The first of these is a simple geometric sum. The second function does not have any simple expression
in terms of standard functions. Indeed it does not converge (as an analytic object) for any z complex z
other than 0. However, we can still manipulate it as a purely formal algebraic object.
We only care about convergence in that we require that that the coefficient of any finite power of z
is finite. This brings us to even more notation:
Definition. Let [z n ]f (z) denote the coefficient of z n in the power series expansion of f (z)
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
!
∞
Version of: 11-Dec-09
X
n
n
n
[z ]f (z) = [z ]
fn z
= fn
n=0
2
2.1
Manipulating formal power series
Definitions
Let a0 , a1 , a2 , . . . be a sequence of real numbers. We call the (possibly infinite) sum a0 + a1 x + a2 x2 +
· · · + ak xk + . . . a formal power series. Here x is a variable, but we do not apriori care if the sum
makes sense for any value of x aside from 0. Despite this, we can consider it a function of x and write
it as A(x). We say that ak is the coefficient of xn in A(x), and we make use of the shorthand
[xk ]A(x) = ak .
The sum is said to be formal because we cannot collapse any of the terms. So, if a0 + a1 x + a2 x2 =
b0 + b1 x + b2 x2 , then it must be that a0 = b0 , a1 = b1 and a2 = b2 . There is a single power series equal to
1: 1 = 1 + 0x + 0x2 + . . . , and a single one equal to 0: 0 + 0x + 0x2 + . . . .
The formal power series turns out to be a very convenient way to store a sequence. We will see
that we can store infinite sequences with a very small amount of memory using what we know about
functions formal power series which arise as Taylor series expansions around 0.
M ARNI M ISHNA , S PRING 2011; K AREN Y EATS, S PRING 2013
MATH 343: A PPLIED D ISCRETE M ATHEMATICS
PAGE 2/6
MATH 895-4 Fall 2010
Course Schedule
L ECTURE 2
f ac ulty of sc ience
depar tment of mathem atics
Week
2.2
Date
Sections
Part/ References
FS2009
Sumfrom
and
Product
Topic/Sections
Generating functions
Notes/Speaker
1
Sept 7 I.1, I.2, I.3
Symbolic methods
A formal
power seriesCombinatorial
is a mathematical
object which behaves essentially
P like nan infinite polynomial.
P
Structures
2
14
I.4,
I.5,
I.6
Unlabelled
structuresseries. Let A(x) =
We can define additionFS:and
of power
Part multiplication
A.1, A.2
n≥0 an x and B(x) =
n≥0 a =
Comtet74
bn x3 n . Then
21
II.1, II.2, II.3
Labelled structures I X
Handout #1
A(x)
+ B(x) :=
(an + bn ) xn
(self study)
4
28
II.4, II.5, II.6
Labelled structures II
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
Combinatorial
parameters
FS A.III
(self-study)
Combinatorial
Parameters
n≥0
X X
Asst #1 Due
A(x) · B(x) :=
ak bn−k xn .
Multivariable GFsn≥0 0≤k≤n
7 Remark,
19
IV.3,
Complex
Analysis
weIV.4have completely
the
coefficient of xn for each n, and
Analytic Methodsspecified
FS:
Partbe
B: IV,
V, VI
are8 well
can
computed
in finite
time given A(x) and B(x).
26 defined. TheyAppendix
Singularity
Analysis
B4
IV.5 V.1
Stanley 99: Ch. 6
9
Nov 2
Asst #2 Due
Asymptotic methods
Handout #1
−1
(self-study)
9
VI.1
Sophie
2.3
so the sum and product
Multiplicative inverse A(x)
10
12
A.3/ C
to Prob. A(x)
Mariolys
The multiplicative
inverse of a formalIntroduction
power series
is the formal power series C(x) =
that satisfies
18
IX.1
Limit Laws and Comb
Marni
11
A(x) · C(x)Sophie
= 1.
Random Structures
20
IX.2
Discrete Limit Laws
Thus,
12
23
IX.3
25
IX.4
and Limit Laws
FS: Part C
(rotating
presentations)
P
n≥0 cn x
n
Combinatorial
XX
Mariolys
instances
of discrete
ak cn−k
xn = 1 + 0x + 0x2 + . . . .
n≥0 k≥n
Continuous Limit Laws
Marni
Recall
that
two formal power seriesQuasi-Powers
are equal
and if and only all of their coefficients are the same. This
13
30
IX.5
Sophie
Gaussian limit laws
leads to the system of equations:
14
Dec 10
Presentations
Asst #3 Due
a0 c0
=
1
(1)
a1 c0 + a0 c1
=
0
(2)
a2 c0 + a1 c1 + a0 c2
=
..
.
0
(3)
(4)
This is a triangular system, and hence we can compute ck once we know of c0 , . . . , ck−1 . We see from (1)
that c0 = 1/a0 , and provided that a0 6= 0, this is well-defined.
Dr.
Marni MISHNA,
Department
of Mathematics,
SIMON
Example:
The
inverse
of A(x)
= 1FRASER
− x. UNIVERSITY
The above
Version of: 11-Dec-09
system simplifies to:
c0
=
1
−c0 + c1
=
0
−c1 + c2
=
..
.
0
−ck + ck+1
=
..
.
0
This has the unique solution: cn = 1 for all n. We should recognize this as the geometric series:
X
−1
(1 − x) =
xn = 1 + x + x2 + x3 + . . . .
n≥0
2.4
Composition
We define composition of formal power series as follows:
X
A(B(x)) :=
an · B(x)n .
n≥0
M ARNI M ISHNA , S PRING 2011; K AREN Y EATS, S PRING 2013
MATH 343: A PPLIED D ISCRETE M ATHEMATICS
PAGE 3/6
MATH 895-4 Fall 2010
Course Schedule
L ECTURE 2
f ac ulty of sc ience
depar tment of mathem atics
Week
2.5
Date
Sections
Part/ References
from FS2009
Differentiation
Topic/Sections
Generating functions
Notes/Speaker
Sept 7 I.1, I.2, I.3
Symbolic methods
Combinatorial
We1 can
develop a complete
theory of derivatives
and integrals of formal power series using the power
Structures
d14 k
k−1
2
I.4,
I.5,
I.6
Unlabelled structures
, and
generalizing
the derivative of a polynomial. Remark, this does not
rule dx x = kx
FS:essentially
Part A.1, A.2
Comtet74
involve
and
rule,
product rule, chain rule) should be proved from
3
21 a limit
II.1, II.2,
II.3 so the usual properties
Labelled (sum
structures
I
Handout #1
the4 definition
of
formal
power
series.
(self
study)
28
II.4, II.5, II.6
Labelled structures II
P
d
n
Combinatorial
Combinatorial
• Definition:
A(x)
= n≥0 (n + 1)a
n+1 x ;
5
Oct 5
III.1, III.2dx
Asst #1 Due
parameters
Parameters
d
d
d FS A.III
6 • Product
12
IV.1,rule:
IV.2
Multivariable
GFs
(A(x)B(x)) = dx
A(x) B(x)
+ A(x) dx
B(x) ;
dx ;(self-study)
7
8
19
IV.3, IV.4
Complex Analysis
d
d
Analytic Methods
(A(x))n = n(A(x))n−1 dx
A(x).
• Chain rule for powers:
FS: Part B:dx
IV, V, VI
26
Singularity Analysis
Appendix B4
IV.5 V.1
Exercise.
Prove
the above
product
and chain rules.
Stanley 99:
Ch. 6
9
Nov 2
10
2.6
9
Asymptotic methods
Handout #1
(self-study)
VI.1
Asst #2 Due
Sophie
TwoA.3/
special
series: exp, Introduction
log
12
C
to Prob.
Mariolys
P
The formal
power
series n≥0 xn /n! isLimit
reminiscent
the Taylor series for the exponential, and hence
18
IX.1
Laws and Comb ofMarni
11
we define
Random Structures
20
IX.2
Discrete Limit Laws X
Sophie
and Limit Laws
exp(x) :=
xn /n!.
12
23
FS: Part C
(rotating
presentations)
IX.3
Combinatorial
Mariolys
instances of discrete n≥0
In fact,
all of the usual properties
of Laws
exp, Marni
but again, since it is defined as the sum, these
25 it satisfies
IX.4
Continuous Limit
properties should be proved from the basic
formal
Quasi-Powers
and power series properties:
13
•
14
30
IX.5
d
exp(x)
dx
Dec 10
Gaussian limit laws
= exp(x);
Presentations
Sophie
Asst #3 Due
• exp(F (x) + G(x)) = exp(F (x)) · exp(G(x));
• exp(F (x))−1 = exp(−F (x)).
Similarly, we can define a series
log(1 − x)−1 :=
X xn
.
n
n≥1
Exercise. Prove the following properties which we can derive from formal power series definitions:
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version
1. of:d11-Dec-09
log(1 − x)−1 = (1 − x)−1 ;
dx
2. log(exp(A(x))) = A(x);
3. exp(log(A(x))) = A(x), if a0 =1;
4. log(A(x) + B(x)) = log(A(x)) + log(B(x)), if a0 = b0 = 1.
3
The basic toolbox for coefficient extraction
There are several key theorems to aid with coefficient extraction. If A(z) =
an .
P∞
n=0
an z n , then [z n ]A(z) :=
Power rule
[z n ]A(pz) = pn [z n ]A(z)
Reduction rule
[z n ]z m A(z) = [z n−m ]A(z)
Sum rule
[z n ] (A(z) + B(z)) = [z n ]A(z) + [z n ]B(z)
M ARNI M ISHNA , S PRING 2011; K AREN Y EATS, S PRING 2013
MATH 343: A PPLIED D ISCRETE M ATHEMATICS
PAGE 4/6
MATH 895-4 Fall 2010
Course Schedule
L ECTURE 2
f ac ulty of sc ience
depar tment of mathem atics
Week
Date
Sections
from FS2009
Product rule
1
Sept 7
I.1, I.2, I.3
2
14
I.4, I.5, I.6
3
21
II.1, II.2, II.3
Binomial
theorem
4
28
II.4, II.5, II.6
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
Part/ References
n
Combinatorial[z
Structures
FS: Part A.1, A.2
Comtet74
Let
n and
k
Handout
#1
(self study)
Combinatorial
parameters
FS A.III
(self-study)
Topic/Sections
n
X
methods
]A(z)Symbolic
× B(z)
=
Generating functions
Notes/Speaker
[z k ]A(z) × [z n−k ]B(z)
k=0
Unlabelled structures
Labelled structures I
be positive
integers.
Labelled structures II r
r!
Combinatorial
.
[z n ](1 + z)r = Asst #1
=Due
(r − n)!n!
n
Parameters
Multivariable GFs
Extended binomial theorem Let k be a positive integer, and r be a real number. Then define
7
19
IV.3, IV.4
Complex Analysis
Analytic Methods
FS: Part B: IV, V, VI
r
r(r
− 1)(r − 2) . . . (r − k + 1)
8
26
Singularity
Analysis
Appendix B4
.
:=
IV.5 V.1
Stanley 99: Ch. 6
k
9
Nov 2
Asst k!
#2 Due
10
11
A
9
Then
12
A.3/ C
18
IX.1
20
Useful
12
VI.1
23
25
Asymptotic methods
Handout #1
(self-study)
Sophie
r
[z ](1 + z) =
.
k
Limit Laws and Comb
Marni
k Prob.
Introduction to
r
Mariolys
Random Structures
IX.2
Discrete Limit Laws
Sophie
Binomial
Formula
and Limit Laws
FS: Part C
Combinatorial
IX.3
−r
(−r)(−r − 1) Mariolys
. . . (−r − k + 1)
(rotating
instances of discrete
=
presentations) k
k!
IX.4
Continuous Limit
Laws Marni
r+k−1
k r+k−1
k and
Quasi-Powers
.
= (−1)
IX.5
Sophie = (−1)
Gaussian limit lawsr − 1
k
13
30
14
Dec 10
Presentations
Asst #3 Due
Example (Catalan numbers).
√
The generating function 1− 2z1−4z is a “famous” generating function. The coefficients are called Catalan
numbers, and we will revisit them shortly as a fundamental counting sequence.
√
1 − 4z
1
n 1−
[z ]
= − (−4)n [z n ](1 + z)1/2
2z
2
1
n 1/2
= − (−4)
2
n
n
−(−4)
1/2(1/2
− 1)(1/2 − 2) . . . (1/2 − n + 1)
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
=
Version of: 11-Dec-09
2
n!
−(−4)n
n
n−1 1 · 1 · 3 · 5 . . . (2n − 3)
=
(1/2) (−1)
2
n!
n−1 1 · 1 · 3 · 5 . . . (2n − 3) (n − 1)!
=2
·
n!
(n − 1)!
| {z }
=1
2n−1 (n−1)!
z
}|
{
1 · 1 · 3 · 5 · · · (2n − 3) · (2 · n) · (2 · (n − 1)) · (2 · 1)
=
n!(n − 1)!
(2n − 2)!
=
(n − 1)!(n − 1)!n
2n − 1 1
=
n−1 n
We will encounter this generating function more than once in this class. This generating function,
as we shall see, is ubiquitous in combinatorics.
Exercise. Go to the web page: http://oeis.org/ This is the Online Encyclopedia of Integer sequences.
The first few terms of this sequence are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862. Enter them into the
search engine and find some combinatorial interpretations of these numbers.
M ARNI M ISHNA , S PRING 2011; K AREN Y EATS, S PRING 2013
MATH 343: A PPLIED D ISCRETE M ATHEMATICS
PAGE 5/6
f ac ulty of sc ience
depar tment of mathem atics
Week
4
Date
Sections
Topic/Sections
Generating functions
Notes/Speaker
from FS2009
A sampling
of counting sequences and generating functions
1
Sept 7
I.1, I.2, I.3
2
14
I.4, I.5, I.6
3
21
II.1, II.2, II.3
4
28
II.4, II.5, II.6
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
4.1
7
Part/ References
MATH 895-4 Fall 2010
Course Schedule
L ECTURE 2
Symbolic methods
Combinatorial
A
Structures
Unlabelled structures
FS: PartPermutations
A.1, A.2
Comtet74
Labelled structures I
HandoutAll
#1 simple, labelled graphs
(self study)
Labelled structures II
Binary words
Combinatorial
Combinatorial
Balanced
parentheses
parameters
Parameters
An
n!
n
2( 2 )
2n
2n
1
Asst #1nDue
n+1
FS A.III
(self-study)
Multivariable GFs
Analytic Methods
Complex Analysis
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
Discrete Limit Laws
Sophie
Combinatorial
instances of discrete
Mariolys
Continuous Limit Laws
Marni
Quasi-Powers and
Gaussian limit laws
Sophie
Advanced
Ideas
19
IV.3, IV.4
A(z)
P
n!z n
P (n) n
2 2 z
1
1−2z
√
1− 1−4x
2
FS: Part B: IV,The
V, VI values for which A(x) is not defined can give us much information
Analysis.
8 • Asymptotic
26
Singularity Analysis
P n n
Appendix B4
1
IV.5 V.1
about
the
asymptotic
growth
of the coefficient. Consider
A(x) = 1−2x
=
2 x . The coefficient
Stanley 99:
Ch. 6
9
Nov 2
Asst
#2
Due
Asymptotic methods
#1
an grows like 2n .Handout
Remark
that
A(x)
is
not
defined
at
x
=
1/2.
In
fact,
very
often
we can formalize
(self-study)
9
VI.1
Sophie
a
connection
between
the
smallest
singularity
of
the
formal
power
series
ρ
(roughly,
the smallest
10
12
A.3/ C
Introduction
Prob.growth
Mariolys
value
for
which A(x) is not defined)
and tothe
of an : an ≈ 1/ρn . Again, complex analysis is
fundamental
in this remarkable Limit
theory.
18
IX.1
Laws and Comb
Marni
11
12
20
IX.2
23
IX.3
25
IX.4
13
30
14
Dec 10
IX.5
Presentations
Asst #3 Due
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
M ARNI M ISHNA , S PRING 2011; K AREN Y EATS, S PRING 2013
MATH 343: A PPLIED D ISCRETE M ATHEMATICS
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