I'm looking for a closed-form formula for the number of partitions of integer $n$ into integer parts less than or equal to 9. Thanks.


closed as off-topic by Travis, colormegone, zz20s, Leucippus, Daniel W. Farlow Apr 22 '16 at 14:57

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    $\begingroup$ PLease give more details. i.e. the context of the question, so that we can see if there's another way than painstakingly calculating this. P.S. I second @N.S.JOHN $\endgroup$ – SinTan1729 Apr 22 '16 at 14:24
  • $\begingroup$ Partitions of integer. $\endgroup$ – user75619 Apr 22 '16 at 14:31
  • $\begingroup$ You mean non-zero partitions. Right? $\endgroup$ – SinTan1729 Apr 22 '16 at 14:32
  • $\begingroup$ Stars and Bars Theorem : This might be what you're looking for. You can easily adjust it for the partitions to be non-zero. $\endgroup$ – SinTan1729 Apr 22 '16 at 14:34
  • $\begingroup$ Yes. I'm interested in the number of ways integer $n$ can be represented as sum of other integers greater than 1, but less than or equal to 9. $\endgroup$ – user75619 Apr 22 '16 at 14:34

One can show that the number $p_k(n)$ of partitions of $n$ into exactly $k$ parts is equal to the number of partitions of $n$ in which the largest part has size $k$. So you are looking for a formula for $p_9(k)$. Rubinstein has given an explicit formula for $p_k(n)$ in terms of Bernoulli polynomials, see here. A. Sills has given Rademacher-type formulas for the restricted partition function.

  • $\begingroup$ Thanks. I'll give it a try. I was hoping for a closed-form formula as a function of integer $n$ though. I've consulted with "Integer partitions" by Andrews. In there closed-form formulas for $p_3(k)$ and $p_4(k)$ are derived, and $p_5(k)$ is provided and is asked to be proven. I have a feeling his approach can deliver formula for $p_9(k)$ as well, but it promises to be quite arduous, so I thought someone must have already done that. :) $\endgroup$ – user75619 Apr 22 '16 at 14:39
  • $\begingroup$ Yes, Sills has an explicit formula of Rademacher-type, i.e., like the explicit formula for the unrestricted partition function $p(n)$ by Rademacher (see Andrew's book). $\endgroup$ – Dietrich Burde Apr 22 '16 at 14:42
  • $\begingroup$ Could you me direct towards Sills results? Where do I find them? $\endgroup$ – user75619 Apr 22 '16 at 14:51
  • $\begingroup$ Here is a link, and a free one also here. $\endgroup$ – Dietrich Burde Apr 22 '16 at 15:12
  • $\begingroup$ I actually found this: math.rutgers.edu/~zeilberg/tokhniot/oPARTITIONS1 Formulas in terms of quasi-polynomials for $p_k(n)$ for $k$s up to 60. Only problem is I don't get how to sum quasi-polynomials. For example, for $k=3$ it is suggested one should add these quasi-polynomials: $[[n^2/12+n/2+47/72], [-1/8, 1/8], [-1/9, -1/9, 2/9]]$. Is it something you would happen to know? Thanks. $\endgroup$ – user75619 Apr 22 '16 at 15:32

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