# Combinatorics of symmetric group:S6

This article gives specific information, namely, combinatorics, about a particular group, namely: symmetric group:S6.

View combinatorics of particular groups | View other specific information about symmetric group:S6

This page discusses some of the combinatorics associated with symmetric group:S6, 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 :**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]