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