structural:expansion
Cycle type

The cycles (including the fixed points) of a permutation σ {\displaystyle \sigma } of a set with n elements partition that set; so the lengths of these cycles form an integer partition of n, which is called the cycle type (or sometimes cycle structure or cycle shape) of σ {\displaystyle \sigma }. There is a "1" in the cycle type for every fixed point of σ {\displaystyle \sigma }, a "2" for every transposition, and so on. The cycle type of β = (1 2 5) (3 4) (6 8) (7) {\displaystyle \beta =(1\,2\,5\,)(\,3\,4\,)(6\,8\,)(\,7\,)} is (3, 2, 2, 1). {\displaystyle (3,2,2,1).}

This may also be written in a more compact form as.More precisely, the general form is {\displaystyle }, where α 1, …, α n {\displaystyle \alpha _{1},\ldots,\alpha _{n}} are the numbers of cycles of respective length. The number of permutations of a given cycle type is

The number of cycle types of a set with n elements equals the value of the partition function p (n) {\displaystyle p(n)}.

Polya 's cycle index polynomial is a generating function which counts permutations by their cycle type.