Element structure of symmetric groups

The symmetric group on a set is the group, under multiplication, of permutations of that set. The symmetric group of degree $$n$$ is the symmetric group on a set of size $$n$$. For convenience, we consider the set to be $$\{ 1,2, \dots, n \}$$.

This article discusses the element structure of the symmetric group of degree $$n$$.

Here are links to more detailed information for small values of degree $$n$$.

General result
The cycle type of a permutation on a set of size $$n$$ is defined as the corresponding unordered integer partition of $$n$$ into the sizes of the cycles in the cycle decomposition. For instance, the permutation $$(1,2,3,4,5)(6,7,8)(9,10,11,12)$$ has cycle type $$5 + 3 + 4$$.

It turns out that there is a bijection between the set of conjugacy classes in the symmetric group of degree $$n$$ and the set of unordered integer partitions via the cycle type map, because cycle type determines conjugacy class.

The size of a conjugacy class corresponding to a cycle type with $$a_j$$ parts of size $$j$$, is:

$$\frac{n!}{\prod_j (j)^{a_j}(a_j!)}$$

See also conjugacy class size formula in symmetric group.

Based on number of cycles
We can partition the set $$S_n$$ based on the number of cycles of each permutation. Since the number of cycles depends only on the cycle type, i.e., on the conjugacy class, this descends to a partition of the set of conjugacy classes in $$S_n$$. The number of cycles can be any number $$k$$, with $$1 \le k \le n$$.


 * The number of elements of $$S_n$$ with exactly $$k$$ cycles is denoted $$|s(n,k)|$$ and is termed the unsigned Stirling number of the first kind with parameters $$n$$ and $$k$$. This is in contrast with the Stirling number of the second kind, which is the number of partitions of a set of size $$n$$ into $$k$$ nonempty subsets. The probability of $$k$$ cycles is thus $$|s(n,k)|/n!$$.
 * The expected number of cycles in an element of $$S_n$$ is the harmonic number $$H_n,$$ defined as $$1 + 1/2 + \dots + 1/n$$. This is approximately $$\log n$$ for lrage $$n$$.

Based on number of fixed points
We can partition the set $$S_n$$ based on the number of fixed points of each permutation. Since the number of fixed points depends only on the cycle type, i.e., on the conjugacy class, this descends to a partition of the set of conjugacy classes in $$S_n$$. The number of fixed points can be any number $$r$$, with $$0 \le r \le n$$ and $$r \ne n - 1$$.


 * The number of elements with $$r$$ fixed points is $$\binom{n}{r}D_{n-r}$$ where $$D_{n-r}$$ is the derangement number, i.e., the number of permutations on a set of size $$n - r$$ with no fixed points.
 * The expected number of fixed points is one.

Other kinds of classifications/groupings of elements not invariant under conjugation
All of these require a total ordering on the set on which the group acts. For convenience, we assume a symmetric group of degree $$n$$ acting the usual way on the set $$\{ 1,2,3, \dots, n \}$$.

For most of these, it is useful to think of the permutation as a string of length $$n$$ given by its one-line notation.

Looking at increase/decrease patterns
Here, we take a permutation $$\sigma$$on the set $$\{ 1,2, \dots, n \}$$ and obtain from it a binary string of length $$n - 1$$ with letters $$I$$ and $$D$$ as follows: if the $$(k+1)^{th}$$ letter in the one-line notation of $$\sigma$$ is greater than the $$k^{th}$$ letter, we write an $$I$$ in the $$k^{th}$$ letter of the binary string, and if it is smaller, we write a $$D$$ in the $$k^{th}$$ letter of the binary sequence. The increasing parts are called rises and the decreasing parts are called runs, so this is also called the rise-run pattern.

We can partition the set of permutations based on the rise/run patterns. Essentially, we are looking at the fibers of the map described above from $$S_n$$ to the set $$\{ I, D \}^{n-1}$$ of size $$2^{n-1}$$.

We may further be interested in grouping these fibers further by creating equivalence classes in $$\{ I, D \}^{n-1}$$. One common way of doing so is to look at the number of occurrences of $$I$$ and $$D$$. The number of permutations that map to strings with $$k$$ $$I$$s is termed the Eulerian number $$A(n,k)$$.

The Robinson-Schensted correspondence
The Robinson-Schensted correspondence encodes a permutation using a ordered pair of Young tableaux of size $$n$$ the same shape -- the position tableau and the shape tableau. This establishes a bijection between the set $$S_n$$ and the disjoint union, over all Young diagrams with $$n$$ boxes, of ordered pairs of Young tableaux of that shape. It turns out that for each Young diagram (which corresponds to an unordered integer partition), there is a corresponding representation of $$S_n$$, and this representation has the same degree as the number of Young tableaux whose shape is that Young diagram.

The algorithm used in the correspondence is termed the bumping algorithm (there are many essentially equivalent variants).

The Bruhat ordering
This is an ordering based on a generating set for $$S_n$$, namely, the generators are of the form $$s_i = (i,i+1)$$. There are $$n - 1$$ such generators. The Bruhat length of an element of $$S_n$$ is the length of the shortest word for it in terms of those letters. We also define a partial ordering on all elements of $$S_n$$ based on whether a shortest word for one can be obtained from a shortest word for the other by multiplying on the left and right by $$s_i$$s. The Bruhat ordering is the precise opposite to an entry-wise comparison ordering of the score matrices. The largest element in the Bruhat ordering is the antidiagonal permutation, given by $$(1,n)(2,n-1) \dots $$. its Bruhat length is $$\binom{n}{2}$$.

The partially ordered set we obtain via the Bruhat ordering exhibits two important symmetries: a symmetry corresponding to conjugation by the antidiagonal (which essentially means reversing the order) and a symmetry corresponding to a left-right interchange. There is also an anti-symmetry of the lattice obtained by left multiplication by the antidiagonal element.