I have the following conjecture that I want to believe has some sort of classical result associated to it, but have yet to find any such evidence.
Let $\ell,r\in\mathbb{Z}^+$, and fix $w_1,\ldots,w_{\ell+1}\in\mathbb{Z}$ such that $|w_k|<r$ for all $k=1,\ldots,(\ell+1)$ and $$\sum_{k=1}^{\ell+1}w_k=0.$$ Then letting $\omega=e^{\frac{2\pi i}{r}}$, I claim $$\sum_{\substack{1\leq s_1,\ldots,s_{\ell+1}\leq r,\\ s_j\neq s_k, \forall k\neq j}}\omega^{w_1s_1+w_2s_2+\cdots+w_{\ell+1}s_{\ell+1}}=(-1)^\ell\ell!.$$
That is, what is the weighted sum over all configurations of the $s_i$s? I have started to try to prove this by induction, but moving past the base case has so far been difficult. Any insight into how to proceed or where to look for relevant theory would be much appreciated!