This might be easy, but let's see.

Question 1.If $\mathfrak{S}_n$ is the group of permutations on $[n]$, then is the following true? $$\sum_{\pi\in\mathfrak{S}_n}\prod_{j=1}^n\frac{j}{\pi(1)+\pi(2)+\cdots+\pi(j)}=1.$$

**Update.** After Lucia's answer, I got motivated to ask:

Question 2.Let $z_1,\dots,z_n$ be indeterminates. Is this identity true too? $$\sum_{\pi\in\mathfrak{S}_n}\prod_{j=1}^n\frac{z_j}{z_{\pi(1)}+z_{\pi(2)}+\cdots+z_{\pi(j)}}=1.$$

**Update.** Now that we've an analytic and an algebraic proof, is there a combinatorial argument too? At least for **Question 1**.