Cycle type of a permutation

by Patrick Stevens Jun 15 2016

The cycle type is an invariant of a permutation in the symmetric group.

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.