Given an element of a symmetric group $S_{n}$ on finitely many elements, we may express $σ$ in cycle notation. The cycle type of $σ$ is then a list of the lengths of the cycles in $σ$ , where conventionally we omit length- $1$ cycles from the cycle type. Conventionally we list the lengths in decreasing order, and the list is presented as a comma-separated collection of values.

The concept is well-defined because disjoint cycle notation is unique up to reordering of the cycles.

Examples

The cycle type of the element $(123) (45)$ in $S_{7}$ is $3, 2$ , or (without the conventional omission of the cycles $(6)$ and $(7)$ ) $3, 2, 1, 1$ .
The cycle type of the identity element is the empty list.
The cycle type of a $k$ -cycle is $k$ , the list containing a single element $k$ .

Cycle type of a permutation