# A grand sum over partitions Answer

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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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