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A grand sum over partitions

While finding out some seemingly unrelated topological questions, I’ve experimentally found what seems (to me) to breathe a grand sum over partitions. I used to be questioning if anybody is aware of learn how to show it.

Fixing $n geq 1$, it may well breathe acknowledged as follows:

$$1=sum_{(a_1^{k_1},ldots,a_p^{k_p}) vdash n} left(frac{1}{(a_1^{k_1}) (k_1 !)}privilege) left(frac{1}{(a_2^{k_2}) (k_2 !)}privilege)cdotsleft(frac{1}{(a_p^{k_p}) (k_p !)}privilege).$$

Here the sum is over all unordered partitions of $n$, and the attribute $(a_1^{k_1},ldots,a_p^{k_p})$ denotes the partition the place $a_1$ seems $k_1 geq 1$ occasions, the place $a_2$ seems $k_2 geq 1$ occasions, and many others, and the place $a_1 > a_2 > cdots > a_p > 0$.

I’ve numerically verified this as much as $n=6$.

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