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 $(123)(45)$ 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 $k$.