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I do not know of representations of the symmetric group $S_n$ of dimension $(n-3)!$ or $(n-2)^(n-2)$ (I know only of representations of the symmetric group $S_n$ of dimension $(n-2)!$ or $(n-1)^(n-1)$). This would be needed to interpret this identity using the Koszul duality of cyclic operads. If you just want to prove it analytically, you should write these functions using $\log$ and the Lambert $W$-function.
