If we have $m$ indistinguishable objects how many ways is it possible to put them in $n$ indistinguihable positions? if we have $m$ indistinguishable objects, how many ways is it possible to put them in $n$ indistinguishable  positions? (for 2 cases 1: without empty position allowed 2: empty positions are allowed.)
We already know that if objects were distinct and the positions are undistinguishable, the number is called Stirling number $S(m,n)$ (without empty positions allowed).

Let us call $A_k(n)$ the number of partitions of $n$ identical objects into $k$ identical boxes allowing empty boxes. If we do not allow empty boxes we simply have $A_k(n-k)$ since we first put one object in each box and then partition the rest without any restrictions.
Now, additionally, consider the number $A_{k,l}(n)$ which is the number of partitions of $n$ objects into $k$ boxes that can fit no more than $l$ objects each. This is the number of Young's diagrams of $n$ squares that fit into a $k\times l$ rectangle. So, $A_{k,l}(n)=A_{l,k}(n)$.
To calculate $A_{k,l}(n)$ consider first choosing how many objects to put in the box with the most objects. Let's say we chose to put $j$ objects in that box. To count how many options we have given that choice we simply need $A_{k-1,j}(n-j)$, since we have $j$ fewer objects to put in boxes, one fewer box to put objects in, and we know that none of these boxes has more than $j$ objects. Therefore, we have the following recursion relation:
$$A_{k,l}(n) = \sum_{j=1}^{l} A_{k-1,j}(n-j)\text.$$
Note that $A_k(n)=A_{k,n}(n)$ so we have our answer.
