Editing Permutation (section)

===Cycle type===
<!-- linked from redirects [[Cycle type]], [[Cycle structure]], and [[Cycle shape]] -->
The cycles (including the fixed points) of a permutation <math>\sigma</math> of a set with {{mvar|n}} elements partition that set; so the lengths of these cycles form an [[integer partition]] of {{mvar|n}}, which is called the '''cycle type''' (or sometimes '''cycle structure''' or '''cycle shape''') of <math>\sigma</math>. There is a "1" in the cycle type for every fixed point of <math>\sigma</math>, a "2" for every transposition, and so on. The cycle type of <math>\beta = (1\,2\,5\,)(\,3\,4\,)(6\,8\,)(\,7\,)</math> is <math>(3, 2, 2, 1).</math>

This may also be written in a more compact form as {{math|[1<sup>1</sup>2<sup>2</sup>3<sup>1</sup>]}}.
More precisely, the general form is <math>[1^{\alpha_1}2^{\alpha_2}\dotsm n^{\alpha_n}]</math>, where <math>\alpha_1,\ldots,\alpha_n</math> are the numbers of cycles of respective length. The number of permutations of a given cycle type is<ref>{{citation|last = Sagan|first = Bruce|title = The Symmetric Group|publisher = Springer | date = 2001 | edition = 2 | page = 3}}</ref>
: <math>\frac{n!}{1^{\alpha_1}2^{\alpha_2}\dotsm n^{\alpha_n}\alpha_1!\alpha_2!\dotsm \alpha_n!}</math>.

The number of cycle types of a set with {{mvar|n}} elements equals the value of the [[Partition function (number theory)|partition function]] <math>p(n)</math>.

[[Pólya enumeration theorem|Polya]]'s [[cycle index]] polynomial is a [[generating function]] which counts permutations by their cycle type.