Edited: Given integer $N\geq 0$, let $$I(N):=\Bigl\{(n\_k)\_{k\geq 1}\in {\mathbb N}^\infty \,:\, n\_k\geq 0, \sum\_{k\geq 1}kn\_k = N \Bigr\}$$ be the set of all decompositions of $N$ as a linear combination of $(1, 2, 3,\ldots)$ with nonnegative integer coefficients. Then $$\sum\_{(n\_k)\in I(N)}\prod\_{k\geq 1} \frac{1}{k^{n\_k} n\_k!} = 1.$$
Is there a simple (non Fourier analytic) proof of this identity?



On decompositions of integers as a linear combination of $(1, 2, 3,\ldots)$
If you multiply both sides by $N!$, then you are just enumerating permutations of $1,2,\dots,N$ according to their cycle type. 


