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Given an integer $N ≥ 0$ and an integer $K ≥ 0$, how many tuples $(n_1,\ldots,n_k)$ are there such that $n_i ≥0$ and $\Sigma n_i = N$? In other words, how many way can you "partition" $N$ into $K$ sets such that the sum of the sets adds up to $N$.

For example, $n = 5$, $k =2$ gives 6 tuples: $(5,0) , (4,1) \ldots (0,5)$.

I've come up with this recursive definition:

$P(n,k) =$ # of tuples $(n_1,..,n_k)$ such that $n_i ≥ 0$ and $\Sigma n_i = N$
$P(n,k) = \Sigma_{i=0}^n{P(i,k-1)}$
$P(0,k)=1 $
$P(n,1)=1$

A suggested answer below is that this problem is the Stirling #'s of 2nd kind but I am not sure how to reduce this problem to it. See my response below.

My question is if there's a closed form solution for $P(n,k)$?

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  • $\begingroup$ I would be very unsurprised if this is a duplicate. If someone less busy than me would be happy to have a check for that, that would be great! $\endgroup$
    – Tara B
    May 15, 2012 at 15:15
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    $\begingroup$ @Tara: You’re right: it’s about the umpty-third question involving this problem or a variant. $\endgroup$ May 16, 2012 at 4:06
  • $\begingroup$ @BrianM.Scott: Are there enough of them to make one of those 'generalisations of common questions'? $\endgroup$
    – Tara B
    May 16, 2012 at 8:11
  • $\begingroup$ @Tara: It would be a chore, because while some are very straightforward, like this one, some have fairly ugly boundary conditions. You can get a pretty good idea if you use the site search on stars and bars. $\endgroup$ May 16, 2012 at 8:16
  • $\begingroup$ @BrianM.Scott: Okay, yes, probably too much work. $\endgroup$
    – Tara B
    May 16, 2012 at 8:25

1 Answer 1

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This is a standard stars-and-bars problem; the answer is that $$p(n,k)=\binom{n+k-1}{k-1}=\binom{n+k-1}n=\frac{(n+k-1)!}{(k-1)!\,n!}\;.$$ The explanation in the Wikipedia article cited above is pretty clear. What you’re counting are called compositions of $n$ into $k$ parts; you’ll find more information on them here. (A search of this site for stars and bars will turn up many variations on this type of problem, some with quite detailed answers, but the Wikipedia stars-and-bars article is your best starting point.)

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