Closed form Formula for Partitions of an Integer

Question

I happened across the following question:

Determine the number of ways of putting $m$ indistinguishable balls into $n$ indistinguishable boxes with the restriction that no box is empty.

The obvious answer is $\Pi(m,n)$, where $\Pi$ denotes the partition of the positive integer $m$ into $n$ parts. This is not "satisfying". Is there a closed form formula for this $\Pi(m,n)$?

JM's answer here might be relevant: https://math.stackexchange.com/questions/34318/feeding-real-or-even-complex-numbers-to-the-integer-partition-function-pn — Sidharth Ghoshal, Nov 08 '23 at 15:08

score 5 · Accepted Answer · edited Apr 13 '17 at 12:20

5

No, there is no known closed form for the partition function. Here is the sequence on OEIS, and also the Wolfram MathWorld page on this subject is very thorough.

Although, looking at this MathSE post, I might be wrong ...

edited Apr 13 '17 at 12:20

Community

1

answered Nov 10 '15 at 07:22

Mike Pierce

19,406

Perfect- just what I was looking for. – Alekos Robotis Nov 10 '15 at 07:23
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@Antonios-AlexandrosRobotis, see this question also. – Mike Pierce Nov 10 '15 at 07:24
There is a formula for the number of partitions of a finite set though. The generating function is $e^{e^{x}-1}$. – Joe Moeller Jul 04 '19 at 07:08

Closed form Formula for Partitions of an Integer

1 Answers1