PARTITIONS OF UNITY Contents 1. Introduction. 1 2. Partitions of
Math 412: Number Theory Lecture 12 Partitions
Transcript of Math 412: Number Theory Lecture 12 Partitions
Math 412: Number TheoryLecture 12 Partitions
Gexin Yu
College of William and Mary
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Partition of integers
A partition � of the positive integer n is a non increasing sequence of
positive integers (�1
,�2
, . . . ,�r
) such that �1
+ �2
+ . . .+ �r
= n.
The integers �1
,�2
, . . . ,�r
are called the parts of the partition �.
We may also specify a partition of n with the frequency of the parts,
namely, n = k
1
a
1
+ k
2
a
2
+ . . .+ k
r
a
r
, where a
1
, . . . , ar
are distinct
nonnegative integers in increasing order.
The number of di↵erent partitions of n is dented by p(n), which is
called the partition function.
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Partition of integers
A partition � of the positive integer n is a non increasing sequence of
positive integers (�1
,�2
, . . . ,�r
) such that �1
+ �2
+ . . .+ �r
= n.
The integers �1
,�2
, . . . ,�r
are called the parts of the partition �.
We may also specify a partition of n with the frequency of the parts,
namely, n = k
1
a
1
+ k
2
a
2
+ . . .+ k
r
a
r
, where a
1
, . . . , ar
are distinct
nonnegative integers in increasing order.
The number of di↵erent partitions of n is dented by p(n), which is
called the partition function.
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Partition of integers
A partition � of the positive integer n is a non increasing sequence of
positive integers (�1
,�2
, . . . ,�r
) such that �1
+ �2
+ . . .+ �r
= n.
The integers �1
,�2
, . . . ,�r
are called the parts of the partition �.
We may also specify a partition of n with the frequency of the parts,
namely, n = k
1
a
1
+ k
2
a
2
+ . . .+ k
r
a
r
, where a
1
, . . . , ar
are distinct
nonnegative integers in increasing order.
The number of di↵erent partitions of n is dented by p(n), which is
called the partition function.
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Restricted partitions
We often study the partitions of n with restriction on the parts. We
use p(n|conditions) to count the partitions of n where the parts
satisfy the conditions specified.
Ex: p(7|all parts are at least 2) = 4.
p
S
(n) is the number of partitions of n into parts from S ;
p
D
(n) is the number of partitions of n into distinct parts;
p
m
(n) is the number of partitions of n into parts each � m.
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Restricted partitions
We often study the partitions of n with restriction on the parts. We
use p(n|conditions) to count the partitions of n where the parts
satisfy the conditions specified.
Ex: p(7|all parts are at least 2) = 4.
p
S
(n) is the number of partitions of n into parts from S ;
p
D
(n) is the number of partitions of n into distinct parts;
p
m
(n) is the number of partitions of n into parts each � m.
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Restricted partitions
We often study the partitions of n with restriction on the parts. We
use p(n|conditions) to count the partitions of n where the parts
satisfy the conditions specified.
Ex: p(7|all parts are at least 2) = 4.
p
S
(n) is the number of partitions of n into parts from S ;
p
D
(n) is the number of partitions of n into distinct parts;
p
m
(n) is the number of partitions of n into parts each � m.
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Restricted partitions
We often study the partitions of n with restriction on the parts. We
use p(n|conditions) to count the partitions of n where the parts
satisfy the conditions specified.
Ex: p(7|all parts are at least 2) = 4.
p
S
(n) is the number of partitions of n into parts from S ;
p
D
(n) is the number of partitions of n into distinct parts;
p
m
(n) is the number of partitions of n into parts each � m.
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Restricted partitions
We often study the partitions of n with restriction on the parts. We
use p(n|conditions) to count the partitions of n where the parts
satisfy the conditions specified.
Ex: p(7|all parts are at least 2) = 4.
p
S
(n) is the number of partitions of n into parts from S ;
p
D
(n) is the number of partitions of n into distinct parts;
p
m
(n) is the number of partitions of n into parts each � m.
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Ferrers diagram
In a Ferrers diagram, we depict the partition n = �1
+ �2
+ . . .+ �r
with �1
� �2
� . . . � �r
with a diagram with k rows of dots such
that row j containing �j
dots, and all rows of dots left justified.
Given a partition n = �1
+ �2
+ . . .+ �r
with �1
� �2
� . . . � �r
, we
define a new partition �0with n = �0
1
+ �02
+ . . .+ �0r
, the conjugate
of �, where �0i
equals the number of parts of � that are at least i .
A partition is self-conjugate if it is its own conjugate.
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Ferrers diagram
In a Ferrers diagram, we depict the partition n = �1
+ �2
+ . . .+ �r
with �1
� �2
� . . . � �r
with a diagram with k rows of dots such
that row j containing �j
dots, and all rows of dots left justified.
Given a partition n = �1
+ �2
+ . . .+ �r
with �1
� �2
� . . . � �r
, we
define a new partition �0with n = �0
1
+ �02
+ . . .+ �0r
, the conjugate
of �, where �0i
equals the number of parts of � that are at least i .
A partition is self-conjugate if it is its own conjugate.
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Ferrers diagram
In a Ferrers diagram, we depict the partition n = �1
+ �2
+ . . .+ �r
with �1
� �2
� . . . � �r
with a diagram with k rows of dots such
that row j containing �j
dots, and all rows of dots left justified.
Given a partition n = �1
+ �2
+ . . .+ �r
with �1
� �2
� . . . � �r
, we
define a new partition �0with n = �0
1
+ �02
+ . . .+ �0r
, the conjugate
of �, where �0i
equals the number of parts of � that are at least i .
A partition is self-conjugate if it is its own conjugate.
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Thm: if n is a positive integer, then the number of partitions of n with
largest part r equals the number of partitions of n into r parts.
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Generating functions
The generating function of a sequence a
n
is the power series
A(x) =
P1n=0
a
n
x
n
.
The A(x) is a formal power series, which should read as “when there
are a
n
objects formed with n elements”.
Ex:
P1n=0
x
n
=
1
1�x
Ex: 1 + x
j
+ x
2j
+ . . . = 1
1�x
j
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Generating functions
The generating function of a sequence a
n
is the power series
A(x) =
P1n=0
a
n
x
n
.
The A(x) is a formal power series, which should read as “when there
are a
n
objects formed with n elements”.
Ex:
P1n=0
x
n
=
1
1�x
Ex: 1 + x
j
+ x
2j
+ . . . = 1
1�x
j
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Generating functions
The generating function of a sequence a
n
is the power series
A(x) =
P1n=0
a
n
x
n
.
The A(x) is a formal power series, which should read as “when there
are a
n
objects formed with n elements”.
Ex:
P1n=0
x
n
=
1
1�x
Ex: 1 + x
j
+ x
2j
+ . . . = 1
1�x
j
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Generating functions
The generating function of a sequence a
n
is the power series
A(x) =
P1n=0
a
n
x
n
.
The A(x) is a formal power series, which should read as “when there
are a
n
objects formed with n elements”.
Ex:
P1n=0
x
n
=
1
1�x
Ex: 1 + x
j
+ x
2j
+ . . . = 1
1�x
j
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Operation on generating functions
Let A(x) =
Pn�0
a
n
x
n
and B(x) =
Pn�0
b
n
x
n
be two generating
functions.
Addition: A(x) + B(x) =
Pn�0
(a
n
+ b
n
)x
n
(count by cases)
Multiplication: A(x)B(x) =
Pn�0
(
Pn
k=0
a
n�k
b
k
)x
n
(count by steps).
This is also called convolution of two generating functions.
Let C (x) =
Pn�0
C
n
x
n
. Then
C
n
=
nX
k=0
a
n�k
b
k
if and only if C (x) = A(x)B(x)
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Operation on generating functions
Let A(x) =
Pn�0
a
n
x
n
and B(x) =
Pn�0
b
n
x
n
be two generating
functions.
Addition: A(x) + B(x) =
Pn�0
(a
n
+ b
n
)x
n
(count by cases)
Multiplication: A(x)B(x) =
Pn�0
(
Pn
k=0
a
n�k
b
k
)x
n
(count by steps).
This is also called convolution of two generating functions.
Let C (x) =
Pn�0
C
n
x
n
. Then
C
n
=
nX
k=0
a
n�k
b
k
if and only if C (x) = A(x)B(x)
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Thm: The generating function for p(n) equals
1X
n=0
p(n)x
n
=
1Y
j=1
1
1� x
j
.
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions
Thm: The generating function for p
D
(n) equals
1X
n=0
p
D
(n)x
n
=
1Y
j=1
(1 + x
j
).
Gexin Yu [email protected] Math 412: Number Theory Lecture 12 Partitions