Fix $k,n\geq 1$. I wish to know the number of ways of dividing $n$ up as follows. First find $c_1,c_2,\ldots ,c_k$ such that $c_i\geq 0$ and $\sum_ic_i=n$. Then take each $c_i$ and find $c_{i,1},\ldots ,c_{i,n}$ such that $c_{i,m}\geq 0$ and $\sum_mmc_{i,m}=c_i$. How many ways are there to do this?
