Evaluate $\sum_{\sigma} (2\pi i)^{-n}\oint \frac{f_{\sigma(1)}(u)\dots f_{\sigma_n(1)}(u)}{(u_2 - u_1)\dots (u_n - u_{n-1})}du_1\dots du_n$

Question

In a probability theory paper I found this rather pleasant result:

Theorem 4.1 Let $n \geq 2$ and $f_1, \dots, f_n : \mathbb{C} \to \mathbb{C}$ be meromorphic with possible poles at $\{ \mathfrak{p}_1, \dots, \mathfrak{p}_n \} $ Then we have identity $$ \sum_{\sigma \in S^{cyc}(n)} \frac{1}{(2\pi \mathbf{i})^n }\oint \dots \oint \frac{f_{\sigma(1)}(u_1)\dots f_{\sigma_n(n)}(u_n) }{(u_2 - u_1)\dots (u_n - u_{n-1})}\;du_1 \dots du_n = \frac{1}{2\pi \mathbf{i} }\oint f_1(u)\dots f_n(u) du $$

It is proven with a lengthy induction process. The authors obtain the result they needed. I wondered if there is more conceptual argument (perhaps using divisors) and could this be generalized beyond cyclic permutations.

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Is there a name for when contour integrals simplify under permutation group action in this way?