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I am looking for whether there is any notation for the $k$-partition number of $n$ where the partitions must include some summand $s$.

An example of the kind of notation I am looking for is $P_k^s(n)$. To exemplify what I mean, see this:

$$P_3^2(6) = |\{ (1,2,3), \ (2,2,2) \} | = 2$$

That is, there are only two $3$-partitions of $6$ that have at least one summand being $2$. I don't quite like my notation here, but that's besides my question. Does anybody know if there's any pre-established, if not conventional, notation for this?

EDIT:

Although $P_k^s(n) = P_{k-1}(n-s)$, and although this fact is present in my paper, I need an intermediate expression for notational/proof reasons.

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    $\begingroup$ Isn't that just "any $k-1$ partition of $n-s$"? $\endgroup$
    – Calvin Lin
    Commented Nov 27, 2023 at 16:21
  • $\begingroup$ @CalvinLin Indeed, and I use this fact in my paper. However, for notation/proof reasons, I need an "intermediate" expression that uses different notation. $\endgroup$
    – user110391
    Commented Nov 27, 2023 at 16:47
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    $\begingroup$ (FYI that's helpful information to put in the writeup) I don't think there's any pre-established convention. You can state your own. I would suggest $ P_3(6; 2)$. $\endgroup$
    – Calvin Lin
    Commented Nov 27, 2023 at 16:53
  • $\begingroup$ @CalvinLin I like that suggestion, thanks :) I will edit my question to include what we talked about. $\endgroup$
    – user110391
    Commented Nov 27, 2023 at 16:55

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