# Combinatorics of symmetric group:S3

This article gives specific information, namely, combinatorics, about a particular group, namely: symmetric group:S3.

View combinatorics of particular groups | View other specific information about symmetric group:S3

This page discusses some of the combinatorics associated with symmetric group:S3, that relies specifically on viewing it as a symmetric group on a finite set. For more on the element structure from a group-theoretic perspective, see element structure of symmetric group:S3.

## 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.

In our case :

Partition | Partition in grouped form | Formula calculating size of fiber for | Size of fiber for | Formula calculating size of fiber for | Size of fiber for | Formula calculating size of fiber for , which is precisely the conjugacy class size for that cycle type (product of previous two formulas) | Size of fiber for , which is precisely the conjugacy class size for that cycle type (product of previous two sizes) |
---|---|---|---|---|---|---|---|

1 + 1 + 1 | 1 (3 times) | 1 | 1 | 1 | |||

2 + 1 | 2 (1 time), 1 (1 time) | 3 | 1 | 3 | |||

3 | 3 (1 time) | 1 | 2 | 2 | |||

Total (3 rows -- number of rows equals number of unordered integer partitions of 3) | -- | -- | 5 (equals the size of , termed the Bell number, at ) | -- | 4 (see Oeis:A107107) | -- | 6 (equals 3!, the size of the symmetric group) |

## Young diagrams and tableaux under the Robinson-Schensted correspondence

### Summary

Partition | Number of Young tableaux for that shape | Hook length formula | Number of permutations (via Robinson-Schensted correspondence) equals square of number of Young tableaux | List of permutations (each permutation written using one-line notation) |
---|---|---|---|---|

1 + 1 + 1 | 1 | 1 | ||

2 + 1 | 2 | 4 | , , , | |

3 | 1 | 1 |

Note that the numbers in the first column are also the degrees of irreducible representations, see linear representation theory of symmetric groups and linear representation theory of symmetric group:S3.

### Partition details

`Further information: Robinson-Schensted correspondence for symmetric group:S3`

Here the partition is the partition for the Young diagram under the Robinson-Schensted correspondence, *not* the partition for the cycle type of the permutation.

Permutation (one-line notation) | Partition (Young diagram) | Position tableau | Shape tableau |
---|---|---|---|

1 + 1 + 1 | |||

2 + 1 | |||

2 + 1 | |||

2 + 1 | |||

2 + 1 | |||

3 |

## Ascent/descent patterns

One-line notation for permutation | Ascent/descent pattern (whether each element is greater or smaller than its predecessor; an A denotes ascent, a D denotes decrease) | Number of Is |
---|---|---|

1,2,3 | AA | 2 |

2,1,3 | DA | 1 |

1,3,2 | AD | 1 |

2,3,1 | AD | 1 |

3,1,2 | DA | 1 |

3,2,1 | DD | 0 |