Suppose we have $n$ elements, assume there is a permutations over $k$ elements among the $n$ elements so $n-k$ are fixed. Let that the permutation over the k elements is represented by permutation cycles so the length of all permutation cycles $=k$.

As an example: Suppose we have the following permutation 

$$ x = \left( {\begin{array}{c}
 x\_1 = \left( {\begin{array}{c}
 1 \\
 2 \\ 
 \end{array} } \right) \\
 x\_2 = \left( {\begin{array}{c}
 3 \\
 4 \\ 
 5 \\
 \end{array} } \right) \\
 x\_3 = \left( {\begin{array}{c}
 6 \\
 7 \\ 
 \end{array} } \right) \\ 
 8 \\
 9 \\
 \vdots \\
 15 \\
 \end{array} } \right)$$

My question: What is the number of permutations we can construct from the $n$ elements where each permutation should consists of the same cycles type?

**Addition:** I know that the number of $k-$cycles in the symmetric group $S\_n$ is $\binom{n}{k}(k-1)!$ but I don't know what to do for the constraint asking that each permutation cycle has the same length in all permutations!