how many ways to partition a set with k subsets, each of fixed size? if $(v_1,...,v_k)$ is a partition of $n$, how many ways $M((v_1,...,v_k),n)$ is there to create a set (partition) of $k$ elements, each of size $v_i$ , i=1,...,k from $n \geq k$ distinct elements ?
example 
if n=3, we consider the 3 distinct elements a,b,c
M((1,1,1),3)=1  : {{a},{b},{c}}
M((1,2),3)=3 :  {{a,b},{c}}  ; {{a,c},{b}}   ;  {{b,c},{a}} 
M((3),3)=1 :   {{a,b,c}}

The species here for distinct $v_k$ is $$\prod_{q=1}^k \mathfrak{P}_{=v_k}(\mathcal{Z}).$$
This gives the exponential generating function
$$G(z) = \prod_{q=1}^k \frac{z^{v_k}}{v_k!}.$$
The desired result is $$n! [z^n] G(z) = n! \prod_{q=1}^k \frac{1}{v_k!}$$ as claimed.
