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Let $q_r(n)$ be the number of partitions of the positive integer $n$ allowing at most $r$ repetitions of any of the parts. (For $r=1$ this is just the usual number of partitions of $n$ into distinct parts.) For $n = 5$ and $r = 2$ these partitions are precisely

5, 4 + 1, 3 + 2, 3 + 1 + 1, and 2 + 2 + 1.

So $q_2(5) = 5$.

Is there a formula (explicit or recursive) or any theory at all for $q_2(n)$ or the general $q_r(n)$?

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  • $\begingroup$ Stating the obvious: the generating function of these numbers is $\sum_{n\geq 0} q_r(n) x^n = \prod_{i\geq 1} \frac{1-x^{i(r+1)}}{1-x^i}$. $\endgroup$
    – Sam Hopkins
    Commented Apr 21, 2023 at 22:53
  • $\begingroup$ But it's unclear what you mean by a "formula." For example, what "formula" did you have in mind in the case $r=1$ (or $r=\infty$, for that matter)? $\endgroup$
    – Sam Hopkins
    Commented Apr 21, 2023 at 23:06
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    $\begingroup$ See also: en.wikipedia.org/wiki/Glaisher%27s_theorem. $\endgroup$
    – Sam Hopkins
    Commented Apr 21, 2023 at 23:12
  • $\begingroup$ Thanks for the reference to Glaisher's theorem. $\endgroup$
    – sqd
    Commented Apr 21, 2023 at 23:21
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    $\begingroup$ As OP asks whether there is a recursive formula, I imagine OP would settle for something that does for $q_2(n)$ what Euler's Pentagonal Number Theorem does for $r=\infty$. I would recommend calculating a few values of $q_2(n)$ and then consulting the Online Encyclopedia of Integer Sequences, oeis.org – OK, I took my own advice, it's here: oeis.org/A000726 with many references and links. $\endgroup$
    – Gerry Myerson
    Commented Apr 22, 2023 at 3:18

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