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Given two positive numbers $r,s$, what is a positive solution of $$\sum_{i=0}^{m-1} k_i = r \quad\text{ and } \quad \sum_{i=0}^{m-1} (i+1)k_i = s \quad ??$$

I would like to get some reference to treat this kind of partition.

remark: There is a famous related subject called "Ewens's sampling formula".

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1 Answer 1

This is a partition of $s$ into $r$ non-zero parts, with $k_i$ parts $i+1$. It would seem slightly more straightforward to write it as

$$ \sum_{i=1}^mk_i=r\quad\text{and}\quad\sum_{i=1}^mik_i=s $$

with $k_i$ parts $i$.

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This is so far from an answer. I don't understand why the people are voting up. It is nothing more than an opinion about how to write a problem, without a solution, answer or hint. –  Lewis S. Feb 2 '13 at 14:54
    
@Lewis: Then I think you need to state your question more specifically. It seemed to me from the question that you weren't aware that this is a partition of $s$ into $r$ non-zero parts. Since you can find loads of information on that by googling once you know what it's called, I was under the impression that I had given you the main information that you'd been lacking. If that's not the case, I suggest that you edit the question to reflect more specifically what it is you want to know about these partitions. –  joriki Feb 2 '13 at 16:25

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