Combinatorics of symmetric group:S4
This article gives specific information, namely, combinatorics, about a particular group, namely: symmetric group:S4.
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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 (4 times)||1||1||1|
|2 + 1 + 1||2 (1 time), 1 (2 times)||6||1||6|
|2 + 2||2 (2 times)||3||1||3|
|3 + 1||3 (1 time), 1 (1 time)||4||2||8|
|4||4 (1 time)||1||6||6|
|Total (5 rows -- number of rows equals number of unordered integer partitions of 4)||--||--||15 (equals the size of , termed the Bell number, for )||--||11 (see Oeis:A107107)||--||24 (equals 4!, the size of the symmetric group)|