Math 412: Number Theory Lecture 12 Partitions

Math 412: Number TheoryLecture 12 Partitions

Gexin Yu

[email protected]

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