Given an element $\sigma$ of a symmetric group $S_n$ on finitely many elements, we may express $\sigma$ in cycle notation. The cycle type of $\sigma$ is then a list of the lengths of the cycles in $\sigma$, 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 $\left(123\right)\left(45\right)$ in $S_7$ is $3,2$, or (without the conventional omission of the cycles $\left(6\right)$ and $\left(7\right)$) $3,2,1,1$.

• 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 $k$.