\begin{definition}[Definition:Cycle Type]
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.
\end{definition}