$$ \int_0^L{dx_1 \int_0^{x_1}{dx_2 \cdots \int_0^{x_{n-1}}{dx_n f(|x_i-x_j|)} } } =\frac{1}{n!} \int_0^L{dx_1 \int_0^{L}{dx_2 \cdots \int_0^{L}{dx_n f(|x_i-x_j|)} } } $$ where $f(|x_i-x_j|)$ means a function dependent on all possible $|x_i-x_j|\,,i\ne j$.
What is the condition of $f$ that satisfies this identity? Thanks.