Combinatorics of symmetric group:S4
This article gives specific information, namely, combinatorics, about a particular group, namely: symmetric group:S4.
View combinatorics of particular groups | View other specific information about symmetric group:S4
This page discusses some of the combinatorics associated with symmetric group:S4, that relies specifically on viewing it as a symmetric group on a finite set. For more on the element structure from a group-theoretic perspective, see element structure of symmetric group:S4.
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 ![]() |
Size of fiber for ![]() |
---|---|---|---|---|---|---|---|
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 ![]() ![]() |
-- | 11 (see Oeis:A107107) | -- | 24 (equals 4!, the size of the symmetric group) |