# Cycle type of a permutation

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## Definition

Let $S$ be a finite set and $\sigma:S \to S$ be a permutation. The cycle type of $\sigma$ is the data of how many cycles of each length are present in the cycle decomposition of $\sigma$. There are two typical ways of specifying the cycle type.

The definition also applies for infinite sets; here, we also need to include cycles of infinite length, i.e., the chains.

### Definition as an unordered list of cycle sizes

The cycle type of a permutation is defined as the unordered list of the sizes of the cycles in the cycle decomposition of $\sigma$. For instance, consider the permutation with cycle decomposition:

$(1,3,5)(2,4)(6)(7,8)$

This permutation has cycle type $(3,2,1,2)$. Since this is an unordered list, this can also be written as $(1,2,2,3)$ or $(1,2,3,2)$.

Note that the sum of all the cycle sizes must equal the size of the whole set $S$. Thus, the cycle type of a permutation is an unordered integer partition of the size of the set.

### Definition as information of how many cycles of each length there are

This decsribes the cycle type as an ordered sequence $i_1, i_2, i_3, \dots$ where $i_j$ is the number of cycles of size (length) $j$. Thus, the permutation:

$(1,3,5)(2,4)(6)(7,8)$

has $i_1 = 1, i_2 = 2, i_3 = 1$ and $i_j = 0, j \ge 4$.

Note that give the cycle type in either form, it can be converted to the other form.