We have a set $S$ with $E$ elements of which only $N$ are unique. We of course know how many repetitions of each of the $N$ elements are present: element $s_i$ is repeating $t_i$ times.
I would like to count the number of ways we can divide the $E$ elements in $N$ blocks of size $k_1, k_2, \cdots , k_N$ when the elements within the block are indistinguishable.
If the $E$ elements are all unique, we can answer directly using the Bell Polynomials.
Do you think is it possible to extend the above result ?