Cycle type of a permutation
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 \(k\).
- Cycle notation in symmetric groups
Cycle notation is a convenient way to represent the elements of a symmetric group.