# Cycle type of a permutation

## Definition

Let  be a finite set and  be a permutation. The cycle type of  is the data of how many cycles of each length are present in the cycle decomposition of . 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 . For instance, consider the permutation with cycle decomposition:



This permutation has cycle type . Since this is an unordered list, this can also be written as  or .

Note that the sum of all the cycle sizes must equal the size of the whole set . 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  where  is the number of cycles of size (length) . Thus, the permutation:



has  and .

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