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.
- 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$~$.