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Two Integer Partition Problems

Let $P(n,k,m)$ be the number of partitions of $n$ into $k$ parts with all parts $\leq m$.

So $P(10,3,4) = 2$, i.e., (4,4,2); (4,3,3).

I need help proving the following:

$P(2n,3,n-1) = P(2n-3,3, n-2)$
$P(4n+3, 3, 2n+1) = P(4n,3,2n-1) + n + 1$.

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Have you tried something? If you're interested in the solution perhaps you should tell us what you've tried so far. If this is homework then you should add that to the tags.) –  Patrick Da Silva Jan 18 '12 at 17:14

1 Answer 1

For the first one: For any partition $2n=a+b+c$ where $a,b,c \leq n-1$, we have a partition $2n-3=(a-1)+(b-1)+(c-1)$ where $a-1,b-1,c-1 \leq n-2$ and vice versa.

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To do this you also have to show that $c$ (the smallest part of $2n$) is at least $2$, so that $c-1$ is positive. This is not difficult as $c= n- (a+b) \ge n-((n-1)+(n-1))=2$. –  Henry Jan 18 '12 at 17:40
    
Thank you @Henry. –  James Jan 18 '12 at 17:49

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