# Definition(s) > [!NOTE] Definition (Cycle Type) > Let $n\in \mathbb{N}^+$, $S_{n}$ denote the [[Symmetric Groups of Finite Degree|nth symmetric group]] and $\sigma\in S_{n}$ be a permutation of $n$ letters. Then the cycle type of $\sigma$ is defined by $\prod_{k=1}^\infty k^{r_{k}}=1^{r_{1}}2^{r_{2}}3^{r_{3}}4^{r_{4}}\cdots$ where $r_{k}$ is the number of [[Cyclic Permutation of n Letters|cycles]] of length $k$ in the [[Existence of Disjoint Cycle Decomposition for Permutations of n Letters|disjoint cycle decomposition]] of $\sigma$. **Remark**: some authors write $(r_{1},r_{2},r_{3},\dots)$ instead. > [!Example] Example > Contents # Properties(s) # Application(s) **More examples**: # Reference(s)