# Combinatorics of symmetric group:S5

This article gives specific information, namely, combinatorics, about a particular group, namely: symmetric group:S5.

View combinatorics of particular groups | View other specific information about symmetric group:S5

This page discusses some of the combinatorics associated with symmetric group:S5, that relies specifically on viewing it as a symmetric group on a finite set.

## 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 + 1 | 1 (5 times) | 1 | 1 | 1 | |||

2 + 1 + 1 + 1 | 2 (1 time), 1 (3 times) | 10 | 1 | 10 | |||

3 + 1 + 1 | 3 (1 time), 1 (2 times) | 10 | 2 | 20 | |||

2 + 2 + 1 | 2 (2 times), 1 (1 time) | 15 | 1 | 15 | |||

4 + 1 | 4 (1 time), 1 (1 time) | 5 | 6 | 30 | |||

3 + 2 | 3 (1 time), 2 (1 time) | 10 | 2 | 20 | |||

5 | 5 (1 time) | 1 | 24 | 24 | |||

Total (7 rows -- number of rows equals number of unordered integer partitions of 5) | -- | -- | 52 (equals the size of , termed the Bell number, at ) | -- | 37 (see Oeis:A107107) | -- | 120 (equals 5!, the size of the symmetric group) |