I know there are similar problems already on the board. However, none of the previously stated questions contain problems where limitations are placed on the BOXES.
Thus, seeing that I am struggling with finding answers to these limitation questions, here is a generalization of different cases of combinatorics problems that I have come across.
A general way of arranging M Distinct Objects into N Identical Boxes With Limitations. i.e (Sterling Numbers of The Second Kind With Limitations placed On Boxes).
Case 1:
Distribute M-Distinct balls into N-Identical Boxes where there are atleast K Boxes that are NOT EMPTY.
Example: S(5,3); where there are ATLEAST 3 boxes that are Not Empty.
Case 2:
Distribute M-Distinct balls into N-Identical Boxes where there are atmost K Boxes that are NOT EMPTY.
Example: S(5,3); where there are at most 3 boxes that are Not Empty.
Sterling numbers of the second kind, e.g. S(5,3) count partitions of a set of 5 objects into 3 blocks, where none of the boxes are empty. What you're suggesting has 3 parameters, say S(n,m,k), a partition of a set of n objects into at most m blocks where at least k are used. But that would just be $\sum_{i=k}^{m} S(n,i)$; the possible number of blocks used goes from k to m, and from there, it's just Sterling numbers of the second kind.
Your second case, if I understand correctly, would just be the number of partitions of a set of n objects into at most k parts, in which case the m isn't really relevant unless m < k. If k $\leq$ m, then it's just $\sum_{i=1}^{k} S(n,i)$. If m < k, then the sum just goes from 1 to m. Either way, you never use more than k parts.