# Transform integration regions

$$\int_0^L{dx_1 \int_0^{x_1}{dx_2 \cdots \int_0^{x_{n-1}}{dx_n f(|x_i-x_j|)} } } =\frac{1}{n!} \int_0^L{dx_1 \int_0^{L}{dx_2 \cdots \int_0^{L}{dx_n f(|x_i-x_j|)} } }$$ 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.

• Are you sure it is not $L^n/n!$ instead of $1/n!$ ? – Tom-Tom Dec 14 '17 at 14:35
• @Tom-Tom Sure.. – xiaohuamao Dec 14 '17 at 14:42