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I'm working on a project that involves that set $P = \{\{n_1, \ldots, n_k\} \mid k \in \mathbb{N}, n_i \in \mathbb{N} \text{ and } n_1 + \cdots +n_k = n\}$ of all integer partitions of a number $n$. Is there a standard notation for this?

I know that $p(n)$ is commonly used to denote the number of integer partitions of $n$, but that's not what I'm looking for and I didn't find any standard notation for $P$.

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Often it is simply denoted as $P(n)$ or $\mathcal{P}(n)$. However, I have seen other letters used in books and articles. I would simply go with $\mathcal{P}(n)$ but you should still formally define it in the beginning of your text.

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I would simply overload the $p$ notation to refer to the set of partitions when its argument is a set. That way you have $|p(S)| = p(|S|)$ which is notationally elegant.

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