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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