# Combinatorics of symmetric group:S3

From Groupprops

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.

## 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 |

## Increase/decrease patterns

One-line notation for permutation | Increase/decrease pattern (whether each element is greater or smaller than its predecessor; an I denotes increase, a D denotes decrease | Number of Is |
---|---|---|

1,2,3 | II | 2 |

2,1,3 | DI | 1 |

1,3,2 | ID | 1 |

2,3,1 | ID | 1 |

3,1,2 | DI | 1 |

3,2,1 | DD | 0 |