The stirling number of the second kind $S(n,k)$ counts the number of partitions of the set $[n]$ into $k$ non-empty parts. I found a definition for the numbers called the $r$-associated stirling numbers of the second kind in wikipedia. These count the number of partitions of $[n]$ into $k$ non-empty parts such that all of the parts have at least $r$ elements. They seem to be "not too hard to calculate" via the recursion:

$S_r(n+1,k)=kS_r(n,k)+\binom{n}{r-1}S_r(n-r+1,k-1)$.

I am interested in a very similar construction, for which I have been unable to find any references. I would like to count $F_r(n,k)$ defined as the number of partitions of $[n]$ into $k$ non-empty parts so that each of them has size $r$ or less.

Most of all I am interested in $\sum\limits_{k=0}^nF_r(n,k)$, basically the number of partitions of $[n]$ into parts of size $r$ or less.

Thank you very much in advance

Regards.

I am aware that the problem of finding compositions of $n$ into numbers less than or equal to $k$ has been studied. But I do not think one of them can be used to compute the other.