# Set 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:

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 |