Let $X=\{\underbrace{a_1,\cdots ,a_1}_{\nu_1},\cdots,\underbrace{a_k,\cdots ,a_k}_{\nu_k}\}$ be a multiset of cardinality $\sum{\nu_i}=n$ where each $a_i$ repeats $\nu_i$ times. We suppose that when $\nu_i=\nu_j$, $a_i$ and $a_j$ are indistinguishable. What is the number of possible partitions of $X$ such that no $a_i$ is put in the same partition as $a_j$, $i\not = j$.
Example 1. $X=\{a,a,b\}$ we have $3$ possible partitions :
$(\{a,a\},\{b\})$ , $(\{a\},\{b\},\{a\})$, $(\{b\},\{a,a\})$
Example 2. $X=\{a,a,b,b\}$ we have $3$ possible partitions : $(\{a,a\},\{b,b\})$, $(\{a\},\{b,b\},\{a\})$, $(\{a\},\{b\},\{a\},\{b\})$
Added: i think the answer is ${1\over r}*{n!\over \nu_1!\cdots\nu_k!}$ where $r$ is the number of equal $\nu_i$. is this correct?