# Difference between revisions of "Combinatorics of symmetric group:S3"

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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 $\mathcal{B}(n)$ the set of all unordered set partitions of $\{ 1,2,\dots,n\}$ into subsets and by $P(n)$ the set of unordered integer partitions of $n$. There are natural combinatorial maps:

$S_n \to \mathcal{B}(n) \to P(n)$

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 $\{ 1,2,\dots,n\}$. The second map sends a subset partition to the partition of $n$ given by the sizes of the parts. The composite of the two maps is termed the cycle type, and classifies conjugacy classes in $S_n$, because cycle type determines conjugacy class.

Further, if we define actions as follows:

• $S_n$ acts on itself by conjugation
• $S_n$ acts on $\mathcal{B}(n)$ by moving around the elements and hence changing the subsets
• $S_n$ acts on $P(n)$ trivially

then the maps above are $S_n$-equivariant, i.e., they commute with the $S_n$-action. Moreover, the action on $\mathcal{B}(n)$ is transitive on each fiber above $P(n)$ and the action on $S_n$ is transitive on each fiber above the composite map to $P(n)$. In particular, for two elements of $\mathcal{B}(n)$ that map to the same element of $P(n)$, the fibers above them in $S_n$ have the same size.

There are formulas for calculating the sizes of the fibers at each level.

In our case $n = 3$:

Partition Partition in grouped form Formula calculating size of fiber for $\mathcal{B}(n) \to P(n)$ Size of fiber for $\mathcal{B}(n) \to P(n)$ Formula calculating size of fiber for $S_n \to \mathcal{B}(n)$ Size of fiber for $S_n \to \mathcal{B}(n)$ Formula calculating size of fiber for $S_n \to P(n)$, which is precisely the conjugacy class size for that cycle type (product of previous two formulas) Size of fiber for $S_n \to P(n)$, which is precisely the conjugacy class size for that cycle type (product of previous two sizes)
1 + 1 + 1 1 (3 times) $\frac{3!}{(1!)^33!}$ 1 $(1 - 1)!^3$ 1 $\frac{3!}{(1)^33!}$ 1
2 + 1 2 (1 time), 1 (1 time) $\frac{3!}{(2!)(1!)}$ 3 $(2 - 1)!(1 - 1)!$ 1 $\frac{3!}{(2)(1)}$ 3
3 3 (1 time) $\frac{3!}{3!}$ 1 $(3 - 1)!$ 2 $\frac{3!}{3}$ 2
Total (3 rows -- number of rows equals number of unordered integer partitions of 3) -- -- 5 (equals the size of $\mathcal{B}(n)$, termed the Bell number, at $n = 3$) -- 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 $\frac{3!}{3 \cdot 2 \cdot 1}$ 1 $[1,2,3]$
2 + 1 2 $\! \frac{3!}{3 \cdot 1 \cdot 1}$ 4 $[1,3,2]$, $[3,1,2]$, $[2,1,3]$, $[2,3,1]$
3 1 $\! \frac{3!}{3 \cdot 2 \cdot 1}$ 1 $[3,2,1]$

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
$1,2,3$ 1 + 1 + 1 $\begin{pmatrix}1 \\ 2 \\ 3 \\\end{pmatrix}$ $\begin{pmatrix} 1 \\ 2 \\ 3 \\\end{pmatrix}$
$2,1,3$ 2 + 1 $\begin{pmatrix}1 & 2 \\ 3 & \\\end{pmatrix}$ $\begin{pmatrix}1 & 2 \\ 3 & \\\end{pmatrix}$
$1,3,2$ 2 + 1 $\begin{pmatrix}1 & 3 \\ 2 & \\\end{pmatrix}$ $\begin{pmatrix} 1 & 3 \\ 2 & \\\end{pmatrix}$
$2,3,1$ 2 + 1 $\begin{pmatrix}1 & 2 \\ 3 & \\\end{pmatrix}$ $\begin{pmatrix} 1 & 3 \\ 2 & \\\end{pmatrix}$
$3,1,2$ 2 + 1 $\begin{pmatrix}1 & 3 \\ 2 & \\\end{pmatrix}$ $\begin{pmatrix} 1 & 2 \\ 3 & \\\end{pmatrix}$
$3,2,1$ 3 $\begin{pmatrix} 1 & 2 & 3 \\\end{pmatrix}$ $\begin{pmatrix} 1 & 2 & 3 \\\end{pmatrix}$

## 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