Set of unordered integer partitions
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(Redirected from Number of unordered integer partitions)
Definition
Let be a nonnegative integer. An unordered integer partition of
is an additive partition of
into positive integers, without any specific ordering on the parts. The set of unordered integer partitions of
, sometimes denoted
, is the set of all such unordered integer partitions.
The cardinality of this set, also termed the number of unordered integer partitions or partition number, is denoted . We have:
.
The set of unordered integer partitions figures in the following ways:
- It is in canonical bijection with the set of conjugacy classes in the symmetric group of degree
. The bijection is via the cycle type map, and it is a bijection because cycle type determines conjugacy class.
- It is in canonical bijection with the set of irreducible representations over the rationals (and also, over any algebraic extension of the rationals) of the symmetric group of degree
. Further information: Linear representation theory of symmetric groups
- For any fixed prime number
, it is in canonical bijection with the set of isomorphism classes of abelian groups of order
, via the structure theorem for finitely generated abelian groups.
Examples
We have the following small values:
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List of partitions | Application to conjugacy class structure of symmetric group | Application to irreducible representation structure of symmetric group | Application to abelian groups of prime power order |
---|---|---|---|---|---|
0 | 1 | The empty partition | trivial group has unique conjugacy class | trivial group has unique conjugacy class | only the trivial group |
1 | 1 | The trivial partition ![]() |
trivial group has unique conjugacy class | trivial group has unique conjugacy class | the unique group of prime order, see equivalence of definitions of group of prime order |
2 | 2 | ![]() ![]() |
link | link | classification of groups of prime-square order |
3 | 3 | ![]() ![]() ![]() |
link | link | link |
4 | 5 | ![]() ![]() ![]() ![]() ![]() |
link | link | link |
5 | 7 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
link | link | link |
6 | 11 | Too long to list | link | link | |
7 | 15 | Too long to list | link | link | |
8 | 22 | Too long to list | link | link | |
9 | 30 | Too long to list | |||
10 | 42 | Too long to list | |||
11 | 56 | Too long to list | |||
12 | 77 | Too long to list | |||
13 | 101 | Too long to list | |||
14 | 135 | Too long to list | |||
15 | 176 | Too long to list | |||
16 | 231 | Too long to list | |||
17 | 297 | Too long to list | |||
18 | 385 | Too long to list | |||
19 | 490 | Too long to list | |||
20 | 627 | Too long to list | |||
21 | 792 | Too long to list | |||
22 | 1002 | Too long to list | |||
23 | 1255 | Too long to list | |||
24 | 1575 | Too long to list |