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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?

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The "partitions" tag is a bit misleading, since, if I get you right, you don't want $c_1\geq c_2\geq ...$? – darij grinberg Aug 20 '10 at 14:02
Anyway, generating functions should do the job. Denoting your number by $p_{n,k}$, we have $\left(\sum\limits_{\ell\in\mathbb N}\text{number of partitions of }\ell\cdot T^{\ell}\right)^k=\sum\limits_{n\in\mathbb N}p_{n,k}T^n$. – darij grinberg Aug 20 '10 at 14:05
... and this means $\left(\prod\limits_{u=1}^{\infty}\left(1-T^u\right)\right)^{-k}=\sum\limits_{n\‌​in\mathbb N}p_{n,k}T^n$. – darij grinberg Aug 20 '10 at 14:06
Here is $k=2$: – darij grinberg Aug 20 '10 at 14:08
The "representation theory" tag doesn't seem to be appropriate – Yemon Choi Aug 20 '10 at 23:55

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