# Combinatorics of symmetric groups

This article gives specific information, namely, combinatorics, about a family of groups, namely: symmetric group.

View combinatorics of group families | View other specific information about symmetric group

This page describes some general combinatorics that can be done with the elements of the symmetric groups.

## Particular cases

(equals order of symmetric group) | symmetric group of degree | combinatorics of this group | |
---|---|---|---|

0 | 1 | trivial group | -- |

1 | 1 | trivial group | -- |

2 | 2 | cyclic group:Z2 | -- |

3 | 6 | symmetric group:S3 | combinatorics of symmetric group:S3 |

4 | 24 | symmetric group:S4 | combinatorics of symmetric group:S4 |

5 | 120 | symmetric group:S5 | combinatorics of symmetric group:S5 |

6 | 720 | symmetric group:S6 | combinatorics of symmetric group:S6 |

7 | 5040 | symmetric group:S7 | combinatorics of symmetric group:S7 |

8 | 40320 | symmetric group:S8 | combinatorics of symmetric group:S8 |

9 | 362880 | symmetric group:S9 | combinatorics of symmetric group:S9 |

## Partitions, subset partitions, and cycle decompositions

Denote by the set of all *unordered* set partitions of into subsets and by the set of unordered integer partitions of . There are natural combinatorial maps:

where the first map sends a permutation to the subset partition induced by its cycle decomposition, which is equivalently the decomposition into orbits for the action of the cyclic subgroup generated by that permutation on . The second map sends a subset partition to the partition of given by the sizes of the parts. The composite of the two maps is termed the cycle type, and classifies conjugacy classes in , because cycle type determines conjugacy class.

Further, if we define actions as follows:

- acts on itself by conjugation
- acts on by moving around the elements and hence changing the subsets
- acts on trivially

then the maps above are -equivariant, i.e., they commute with the -action. Moreover, the action on is transitive on each fiber above and the action on is transitive on each fiber above the composite map to . In particular, for two elements of that map to the same element of , the fibers above them in have the same size.

There are formulas for calculating the sizes of the fibers at each level.