integer partitions - school of mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 ·...
TRANSCRIPT
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Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Integer Partitions
George Kinnear
June 11, 2009
George Kinnear Integer Partitions
![Page 2: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/2.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
1 Partitions
2 Partition Identities
3 The Rogers-Ramanujan Identities
George Kinnear Integer Partitions
![Page 3: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/3.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Definition
Definition
A partition of a positive integer n is a way of writing n as asum of positive integers.
The summands of the partition are known as parts.
Example
4 = 4
= 3 + 1
= 2 + 2
= 2 + 1 + 1
= 1 + 1 + 1 + 1
George Kinnear Integer Partitions
![Page 4: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/4.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Definition
Definition
A partition of a positive integer n is a way of writing n as asum of positive integers.The summands of the partition are known as parts.
Example
4 = 4
= 3 + 1
= 2 + 2
= 2 + 1 + 1
= 1 + 1 + 1 + 1
George Kinnear Integer Partitions
![Page 5: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/5.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Definition
Definition
A partition of a positive integer n is a way of writing n as asum of positive integers.The summands of the partition are known as parts.
Example
4 = 4
= 3 + 1
= 2 + 2
= 2 + 1 + 1
= 1 + 1 + 1 + 1
George Kinnear Integer Partitions
![Page 6: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/6.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Example: Permutations of n objects
Cycle notation: e.g. (1 2), (1 2 3).
Conjugacy classes of Sn are labelled by cycle type:
(1 2 3) cycle type 3
(1 2)(5 7 8) cycle type 2, 3.
Example (S4)
Conjugacy class rep.
Sum of cycle lengths
e
(1)(2)(3)(4) 1 + 1 + 1 + 1
(1 2)
(1 2)(3)(4) 2 + 1 + 1
(1 2 3)
(1 2 3)(4) 3 + 1
(1 2)(3 4)
(1 2)(3 4) 2 + 2
(1 2 3 4)
(1 2 3 4) 4
George Kinnear Integer Partitions
![Page 7: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/7.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Example: Permutations of n objects
Cycle notation: e.g. (1 2), (1 2 3).Conjugacy classes of Sn are labelled by cycle type:
(1 2 3) cycle type 3
(1 2)(5 7 8) cycle type 2, 3.
Example (S4)
Conjugacy class rep.
Sum of cycle lengths
e
(1)(2)(3)(4) 1 + 1 + 1 + 1
(1 2)
(1 2)(3)(4) 2 + 1 + 1
(1 2 3)
(1 2 3)(4) 3 + 1
(1 2)(3 4)
(1 2)(3 4) 2 + 2
(1 2 3 4)
(1 2 3 4) 4
George Kinnear Integer Partitions
![Page 8: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/8.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Example: Permutations of n objects
Cycle notation: e.g. (1 2), (1 2 3).Conjugacy classes of Sn are labelled by cycle type:
(1 2 3) cycle type 3
(1 2)(5 7 8) cycle type 2, 3.
Example (S4)
Conjugacy class rep.
Sum of cycle lengths
e
(1)(2)(3)(4) 1 + 1 + 1 + 1
(1 2)
(1 2)(3)(4) 2 + 1 + 1
(1 2 3)
(1 2 3)(4) 3 + 1
(1 2)(3 4)
(1 2)(3 4) 2 + 2
(1 2 3 4)
(1 2 3 4) 4
George Kinnear Integer Partitions
![Page 9: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/9.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Example: Permutations of n objects
Cycle notation: e.g. (1 2), (1 2 3).Conjugacy classes of Sn are labelled by cycle type:
(1 2 3) cycle type 3
(1 2)(5 7 8) cycle type 2, 3.
Example (S4)
Conjugacy class rep.
Sum of cycle lengths
e (1)(2)(3)(4)
1 + 1 + 1 + 1
(1 2) (1 2)(3)(4)
2 + 1 + 1
(1 2 3) (1 2 3)(4)
3 + 1
(1 2)(3 4) (1 2)(3 4)
2 + 2
(1 2 3 4) (1 2 3 4)
4
George Kinnear Integer Partitions
![Page 10: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/10.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Example: Permutations of n objects
Cycle notation: e.g. (1 2), (1 2 3).Conjugacy classes of Sn are labelled by cycle type:
(1 2 3) cycle type 3
(1 2)(5 7 8) cycle type 2, 3.
Example (S4)
Conjugacy class rep. Sum of cycle lengths
e (1)(2)(3)(4) 1 + 1 + 1 + 1
(1 2) (1 2)(3)(4) 2 + 1 + 1
(1 2 3) (1 2 3)(4) 3 + 1
(1 2)(3 4) (1 2)(3 4) 2 + 2
(1 2 3 4) (1 2 3 4) 4
George Kinnear Integer Partitions
![Page 11: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/11.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
The partition function
Definition
p(n) denotes the number of partitions of n
n 1 2 3 4 5 6 7 8 9 10
p(n) 1 2 3 5 7 11 15 22 30 42
p(200) = ?
3 972 999 029 388
p(n) = 1π√
2
∞∑k=1
Ak(n)√
k
d
dx
sinh(πk
√23
(x − 1
24
))√x − 1
24
x=n
George Kinnear Integer Partitions
![Page 12: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/12.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
The partition function
Definition
p(n) denotes the number of partitions of n
n 1 2 3 4 5 6 7 8 9 10
p(n) 1 2 3 5 7 11 15 22 30 42
p(200) = ?
3 972 999 029 388
p(n) = 1π√
2
∞∑k=1
Ak(n)√
k
d
dx
sinh(πk
√23
(x − 1
24
))√x − 1
24
x=n
George Kinnear Integer Partitions
![Page 13: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/13.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
The partition function
Definition
p(n) denotes the number of partitions of n
n 1 2 3 4 5 6 7 8 9 10
p(n) 1 2 3 5 7 11 15 22 30 42
p(200) = ?
3 972 999 029 388
p(n) = 1π√
2
∞∑k=1
Ak(n)√
k
d
dx
sinh(πk
√23
(x − 1
24
))√x − 1
24
x=n
George Kinnear Integer Partitions
![Page 14: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/14.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
The partition function
Definition
p(n) denotes the number of partitions of n
n 1 2 3 4 5 6 7 8 9 10
p(n) 1 2 3 5 7 11 15 22 30 42
p(200) =
?
3 972 999 029 388
p(n) = 1π√
2
∞∑k=1
Ak(n)√
k
d
dx
sinh(πk
√23
(x − 1
24
))√x − 1
24
x=n
George Kinnear Integer Partitions
![Page 15: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/15.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
The partition function
Definition
p(n) denotes the number of partitions of n
n 1 2 3 4 5 6 7 8 9 10
p(n) 1 2 3 5 7 11 15 22 30 42
p(200) =
?
3 972 999 029 388
p(n) = 1π√
2
∞∑k=1
Ak(n)√
k
d
dx
sinh(πk
√23
(x − 1
24
))√x − 1
24
x=n
George Kinnear Integer Partitions
![Page 16: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/16.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Open problems
Is p(n) prime for infinitely many n?
Is the value of p(n) even approximately half of the time?
George Kinnear Integer Partitions
![Page 17: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/17.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Open problems
Is p(n) prime for infinitely many n?
Is the value of p(n) even approximately half of the time?
George Kinnear Integer Partitions
![Page 18: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/18.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
The basic idea
Rather than just p(n), we can consider placing conditions onthe kinds of partitions allowed.
Example (Partitions of 4)
4 = 4
= 3 + 1
odd parts
= 2 + 2
= 2 + 1 + 1
= 1 + 1 + 1 + 1
odd parts
p(4 | odd parts) = 2.
George Kinnear Integer Partitions
![Page 19: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/19.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
The basic idea
Rather than just p(n), we can consider placing conditions onthe kinds of partitions allowed.
Example (Partitions of 4)
4 = 4
= 3 + 1
odd parts
= 2 + 2
= 2 + 1 + 1
= 1 + 1 + 1 + 1
odd parts
p(4 | odd parts) = 2.
George Kinnear Integer Partitions
![Page 20: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/20.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
The basic idea
Rather than just p(n), we can consider placing conditions onthe kinds of partitions allowed.
Example (Partitions of 4)
4 = 4
= 3 + 1 odd parts
= 2 + 2
= 2 + 1 + 1
= 1 + 1 + 1 + 1 odd parts
p(4 | odd parts) = 2.
George Kinnear Integer Partitions
![Page 21: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/21.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
The basic idea
Rather than just p(n), we can consider placing conditions onthe kinds of partitions allowed.
Example (Partitions of 4)
4 = 4
= 3 + 1 odd parts
= 2 + 2
= 2 + 1 + 1
= 1 + 1 + 1 + 1 odd parts
p(4 | odd parts) = 2.
George Kinnear Integer Partitions
![Page 22: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/22.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Partition identities
Remarkably, there are many identities of the form
p(n | condition A) = p(n | condition B)
Example (Partitions of 4)
4 = 4
distinct parts
= 3 + 1
distinct parts
= 2 + 2
= 2 + 1 + 1
= 1 + 1 + 1 + 1
p(4 | distinct parts) = 2
= p(4 | odd parts).
George Kinnear Integer Partitions
![Page 23: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/23.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Partition identities
Remarkably, there are many identities of the form
p(n | condition A) = p(n | condition B)
Example (Partitions of 4)
4 = 4
distinct parts
= 3 + 1
distinct parts
= 2 + 2
= 2 + 1 + 1
= 1 + 1 + 1 + 1
p(4 | distinct parts) = 2
= p(4 | odd parts).
George Kinnear Integer Partitions
![Page 24: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/24.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Partition identities
Remarkably, there are many identities of the form
p(n | condition A) = p(n | condition B)
Example (Partitions of 4)
4 = 4 distinct parts
= 3 + 1 distinct parts
= 2 + 2
= 2 + 1 + 1
= 1 + 1 + 1 + 1
p(4 | distinct parts) = 2
= p(4 | odd parts).
George Kinnear Integer Partitions
![Page 25: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/25.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Partition identities
Remarkably, there are many identities of the form
p(n | condition A) = p(n | condition B)
Example (Partitions of 4)
4 = 4 distinct parts
= 3 + 1 distinct parts
= 2 + 2
= 2 + 1 + 1
= 1 + 1 + 1 + 1
p(4 | distinct parts) = 2
= p(4 | odd parts).
George Kinnear Integer Partitions
![Page 26: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/26.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Partition identities
Remarkably, there are many identities of the form
p(n | condition A) = p(n | condition B)
Example (Partitions of 4)
4 = 4 distinct parts
= 3 + 1 distinct parts
= 2 + 2
= 2 + 1 + 1
= 1 + 1 + 1 + 1
p(4 | distinct parts) = 2 = p(4 | odd parts).
George Kinnear Integer Partitions
![Page 27: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/27.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Euler’s identity
Theorem (Euler’s identity)
p(n | odd parts) = p(n | distinct parts)
Proof.
odd parts distinct parts
merge pairs of equal parts
split even parts in half
�
George Kinnear Integer Partitions
![Page 28: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/28.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Euler’s identity
Theorem (Euler’s identity)
p(n | odd parts) = p(n | distinct parts)
Proof.
odd parts distinct parts
merge pairs of equal parts
split even parts in half
�
George Kinnear Integer Partitions
![Page 29: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/29.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Euler’s identity
Theorem (Euler’s identity)
p(n | odd parts) = p(n | distinct parts)
Proof.
odd parts distinct parts
merge pairs of equal parts
split even parts in half
�
George Kinnear Integer Partitions
![Page 30: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/30.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Euler’s identity
Theorem (Euler’s identity)
p(n | odd parts) = p(n | distinct parts)
Proof.
odd parts distinct parts
merge pairs of equal parts
split even parts in half
�
George Kinnear Integer Partitions
![Page 31: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/31.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Generating functions
(q1 + q3)(q2 + q4)
= q1+2 + q1+4 + q3+2 + q3+4
= q3 + 2q5 + q7
∞∑n=1
p(n | condition )qn = f(q)
Example ∞∑n=0
p(n | distinct parts )qn =
∞∏n=1
(1 + qn)
∞∑n=0
p(n | odd parts )qn =
∞∏n=1n odd
1
(1 − qn)
George Kinnear Integer Partitions
![Page 32: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/32.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Generating functions
(q1 + q3)(q2 + q4) = q1+2 + q1+4 + q3+2 + q3+4
= q3 + 2q5 + q7
∞∑n=1
p(n | condition )qn = f(q)
Example ∞∑n=0
p(n | distinct parts )qn =
∞∏n=1
(1 + qn)
∞∑n=0
p(n | odd parts )qn =
∞∏n=1n odd
1
(1 − qn)
George Kinnear Integer Partitions
![Page 33: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/33.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Generating functions
(q1 + q3)(q2 + q4) = q1+2 + q1+4 + q3+2 + q3+4
= q3 + 2q5 + q7
∞∑n=1
p(n | condition )qn = f(q)
Example ∞∑n=0
p(n | distinct parts )qn =
∞∏n=1
(1 + qn)
∞∑n=0
p(n | odd parts )qn =
∞∏n=1n odd
1
(1 − qn)
George Kinnear Integer Partitions
![Page 34: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/34.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Generating functions
(q1 + q3)(q2 + q4) = q1+2 + q1+4 + q3+2 + q3+4
= q3 + 2q5 + q7
∞∑n=1
p(n | condition )qn = f(q)
Example ∞∑n=0
p(n | distinct parts )qn =
∞∏n=1
(1 + qn)
∞∑n=0
p(n | odd parts )qn =
∞∏n=1n odd
1
(1 − qn)
George Kinnear Integer Partitions
![Page 35: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/35.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Generating functions
So Euler’s identity is equivalent to
∞∏n=1
(1 + qn) =
∞∏n=1n odd
1
(1 − qn)
This is easily verified:
∞∏n=1
(1 + qn)
=
∞∏n=1
(1 − q2n)
(1 − qn)
=
∞∏n=1n odd
1
(1 − qn)
George Kinnear Integer Partitions
![Page 36: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/36.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Generating functions
So Euler’s identity is equivalent to
∞∏n=1
(1 + qn) =
∞∏n=1n odd
1
(1 − qn)
This is easily verified:
∞∏n=1
(1 + qn)
=
∞∏n=1
(1 − q2n)
(1 − qn)
=
∞∏n=1n odd
1
(1 − qn)
George Kinnear Integer Partitions
![Page 37: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/37.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Generating functions
So Euler’s identity is equivalent to
∞∏n=1
(1 + qn) =
∞∏n=1n odd
1
(1 − qn)
This is easily verified:
∞∏n=1
(1 + qn) =
∞∏n=1
(1 − q2n)
(1 − qn)
=
∞∏n=1n odd
1
(1 − qn)
George Kinnear Integer Partitions
![Page 38: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/38.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
Generating functions
So Euler’s identity is equivalent to
∞∏n=1
(1 + qn) =
∞∏n=1n odd
1
(1 − qn)
This is easily verified:
∞∏n=1
(1 + qn) =
∞∏n=1
(1 − q2n)
(1 − qn)
=
∞∏n=1n odd
1
(1 − qn)
George Kinnear Integer Partitions
![Page 39: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/39.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
The Rogers-Ramanujan identities
p(n | parts ≡ 1 or 4 (mod 5)) = p(n | 2-distinct parts)
p(n | parts ≡ 2 or 3 (mod 5)) = p(n | 2-distinct parts, each > 2)
There is no known direct bijective proof (bijections can beconstructed iteratively, though).
In terms of generating functions:
1 +
∞∑m=1
qm2
(1 − q)(1 − q2) · · · (1 − qm)=
∞∏n=1
1
(1 − q5n−4)(1 − q5n−1)
1 +
∞∑m=1
qm2+m
(1 − q)(1 − q2) · · · (1 − qm)=
∞∏n=1
1
(1 − q5n−3)(1 − q5n−2)
George Kinnear Integer Partitions
![Page 40: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/40.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
The Rogers-Ramanujan identities
p(n | parts ≡ 1 or 4 (mod 5)) = p(n | 2-distinct parts)
p(n | parts ≡ 2 or 3 (mod 5)) = p(n | 2-distinct parts, each > 2)
There is no known direct bijective proof (bijections can beconstructed iteratively, though).
In terms of generating functions:
1 +
∞∑m=1
qm2
(1 − q)(1 − q2) · · · (1 − qm)=
∞∏n=1
1
(1 − q5n−4)(1 − q5n−1)
1 +
∞∑m=1
qm2+m
(1 − q)(1 − q2) · · · (1 − qm)=
∞∏n=1
1
(1 − q5n−3)(1 − q5n−2)
George Kinnear Integer Partitions
![Page 41: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/41.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
The Rogers-Ramanujan identities
p(n | parts ≡ 1 or 4 (mod 5)) = p(n | 2-distinct parts)
p(n | parts ≡ 2 or 3 (mod 5)) = p(n | 2-distinct parts, each > 2)
There is no known direct bijective proof (bijections can beconstructed iteratively, though).
In terms of generating functions:
1 +
∞∑m=1
qm2
(1 − q)(1 − q2) · · · (1 − qm)=
∞∏n=1
1
(1 − q5n−4)(1 − q5n−1)
1 +
∞∑m=1
qm2+m
(1 − q)(1 − q2) · · · (1 − qm)=
∞∏n=1
1
(1 − q5n−3)(1 − q5n−2)
George Kinnear Integer Partitions
![Page 42: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/42.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
History
Discovered in 1894 by L. J. Rogers (1862-1933).
Rediscovered in 1912 by S. Ramanujan (1887-1920), withoutproof.
George Kinnear Integer Partitions
![Page 43: Integer Partitions - School of Mathematics › gkinnear › files › colloq1.pdf · 2011-12-20 · OutlinePartitionsPartition IdentitiesThe Rogers-Ramanujan Identities Definition](https://reader033.vdocuments.site/reader033/viewer/2022060322/5f0d96a17e708231d43b19af/html5/thumbnails/43.jpg)
Outline Partitions Partition Identities The Rogers-Ramanujan Identities
History
Discovered in 1894 by L. J. Rogers (1862-1933).
Rediscovered in 1912 by S. Ramanujan (1887-1920), withoutproof.
George Kinnear Integer Partitions