Combinatorics of symmetric group:S3
This article gives specific information, namely, combinatorics, about a particular group, namely: symmetric group:S3.
View combinatorics of particular groups | View other specific information about symmetric group:S3
This page discusses some of the combinatorics associated with symmetric group:S3, 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 ![]() |
Size of fiber for ![]() |
---|---|---|---|---|---|---|---|
1 + 1 + 1 | 1 (3 times) | ![]() |
1 | ![]() |
1 | ![]() |
1 |
2 + 1 | 2 (1 time), 1 (1 time) | ![]() |
3 | ![]() |
1 | ![]() |
3 |
3 | 3 (1 time) | ![]() |
1 | ![]() |
2 | ![]() |
2 |
Total (3 rows -- number of rows equals number of unordered integer partitions of 3) | -- | -- | 5 (equals the size of ![]() ![]() |
-- | 4 (see Oeis:A107107) | -- | 6 (equals 3!, the size of the symmetric group) |
Young diagrams and tableaux under the Robinson-Schensted correspondence
Summary
Partition | Number of Young tableaux for that shape | Hook length formula | Number of permutations (via Robinson-Schensted correspondence) equals square of number of Young tableaux | List of permutations (each permutation written using one-line notation) |
---|---|---|---|---|
1 + 1 + 1 | 1 | ![]() |
1 | ![]() |
2 + 1 | 2 | ![]() |
4 | ![]() ![]() ![]() ![]() |
3 | 1 | ![]() |
1 | ![]() |
Note that the numbers in the first column are also the degrees of irreducible representations, see linear representation theory of symmetric groups and linear representation theory of symmetric group:S3.
Partition details
Further information: Robinson-Schensted correspondence for symmetric group:S3
Here the partition is the partition for the Young diagram under the Robinson-Schensted correspondence, not the partition for the cycle type of the permutation.
Permutation (one-line notation) | Partition (Young diagram) | Position tableau | Shape tableau |
---|---|---|---|
![]() |
1 + 1 + 1 | ![]() |
![]() |
![]() |
2 + 1 | ![]() |
![]() |
![]() |
2 + 1 | ![]() |
![]() |
![]() |
2 + 1 | ![]() |
![]() |
![]() |
2 + 1 | ![]() |
![]() |
![]() |
3 | ![]() |
![]() |
Ascent/descent patterns
One-line notation for permutation | Ascent/descent pattern (whether each element is greater or smaller than its predecessor; an A denotes ascent, a D denotes decrease) | Number of Is |
---|---|---|
1,2,3 | AA | 2 |
2,1,3 | DA | 1 |
1,3,2 | AD | 1 |
2,3,1 | AD | 1 |
3,1,2 | DA | 1 |
3,2,1 | DD | 0 |