# 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.

# 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 $$k$$.

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