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}}

If the $v_i$ are distinct then the answer is: $$\frac{n!}{v_1!\times\cdots\times v_k!}$$
More general if you have $n=n_1w_1+\cdots+n_mw_m$ with $n_i\in\mathbb Z_{>0}$, distinct $w_i$ and $w_i$ occurring exactly $n_i$ times in tuple $(v_1,\cdots,v_k)$ then multiple counting must be 'corrected' and the answer is:$$\frac{n!}{(w_1!)^{n_1}\times\cdots\times (w_m!)^{n_m}\times n_1!\times\cdots\times n_m!}$$
Here $k=n_1+\cdots +n_m$.
