# Definition:Cycle Type

## Definition

Let $S_n$ denote the symmetric group on $n$ letters.

Let $\rho \in S_n$.

From Existence and Uniqueness of Cycle Decomposition, every $\rho$ may be uniquely expressed as a product of disjoint cycles:

$\rho = \tau_1, \tau_2, \ldots, \tau_r$

up to the order of factors.

Let $\tau_1, \tau_2, \ldots, \tau_r$ be arranged in increasing order of cycle length.

Let the length of the cycle $\tau_i$ be $k_i$.

The resulting ordered tuple of cycle lengths $\tuple {k_1, k_2, \ldots, k_r}$ is called the cycle type of $\rho$.

Thus $\sigma$ and $\rho$ have the same cycle type if they have the same number of cycles of equal length.

## Also known as

Some sources refer to the cycle type of a permutation as its form.