For questions about functions which are symmetric in their arguments.
A function $f : X^n \to Y$ is said to be symmetric if $f(x_{\sigma(1)}, \dots, x_{\sigma(n)}) = f(x_1, \dots, x_n)$ for every $\sigma \in S_n$.
The $k^{\text{th}}$ elementary symmetric polynomial in the variables $x_1, \dots, x_n$ is
$$e_k(x_1, \dots, x_n) = \sum_{1 \leq j_1 < \dots < j_k \leq n}x_{j_1}\dots x_{j_k}.$$
For example, $e_0(x_1, \dots, x_n) = 1$, $e_1(x_1, \dots, x_n) = x_1 + \dots + x_n$, $e_n(x_1, \dots, x_n) = x_1\dots x_n$. Every symmetric polynomial can be written as a linear combination of elementary symmetric polynomials.
