The origin of the question is from Chapter 1, Problem 6.9 in Theory of Point Estimation by Lehmann.
Let $e_k(X_1, \ldots, X_n)$ denote the k-th elementary polynomial. Consider two tuples $(x_1, ..., x_n)$ and $(y_1, ..., y_n)$. If we have $e_k(x_1, \ldots, x_n) = e_k(y_1, \ldots, y_n)$ for $1 \leq k \leq n$, then this implies that the $y$'s are a permutation of the $x$'s.
I was wondering if anyone had a reference (or a hint) to this result. My original idea was to use induction on the number of variables; however, it was not clear to me on how to reduce the induction step and use the induction assumption in a meaningful way that would yield the desired result.