I appreciate if you can help me with a math problem.
While I know the number of partitions of $\{1,2, \dots , n\}$ into $k$ nonempty subsets is "Stirling numbers of the second kind", $S(n,k)$; I want to calculate the number of partitions of $\{1,2, \dots , n\}$ into $k$ nonempty subsets with known length for each subset.
For instance, the number of partitions of $\{1,2,3,4\}$ into $2$ nonempty subsets is $S(4,2)=7$; however, with the length of the subsets equal to $2$ and $2$, this number becomes $C(4,2)/2=3$ or with the length of the subsets equal to $3$ and $1$, this number becomes $C(4,1)=4$.
I appreciate if you inform me of any systematic way or formula to calculate this number for a general case of $n, k$ and $\{L_1,L_2,\dots,L_k\}$ while $L_k$ is the length for the $k-$th subset and it is obvious that sum$(L_1,L_2,\dots,L_k)=n$.
Thank you.