While studying some seemingly unrelated topological questions, I have experimentally discovered what appears (to me) to be a remarkable sum over partitions. I was wondering if anyone knows how to prove it.

Fixing $n \geq 1$, it can be stated as follows:

$$1=\sum_{(a_1^{k_1},\ldots,a_p^{k_p}) \vdash n} \left(\frac{1}{(a_1^{k_1}) (k_1 !)}\right) \left(\frac{1}{(a_2^{k_2}) (k_2 !)}\right)\cdots\left(\frac{1}{(a_p^{k_p}) (k_p !)}\right).$$ Here the sum is over all unordered partitions of $n$, and the symbol $(a_1^{k_1},\ldots,a_p^{k_p})$ denotes the partition where $a_1$ appears $k_1 \geq 1$ times, where $a_2$ appears $k_2 \geq 1$ times, etc, and where $a_1 > a_2 > \cdots > a_p > 0$.

I have numerically verified this up to $n=6$.