# 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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