Symmetric sum

A symmetric sum is any sum in which any permutation of the variables leaves the sum unchanged.

One way to generate symmetric sums is using symmetric sum notation. If $f(x_1, x_2, x_3, \dots, x_n)$ is a function of $n$ variables then the symmetric sum $\sum_{sym} f(x_1, x_2, x_3, \dots, x_n) = \sum_{\sigma} f(x_{\sigma(1)}, {x_\sigma(2)}, {x_\sigma(3)}, \dots, x_{\sigma(n)})$ , where $\sigma$ ranges over all permutations of $(1, 2, 3, \dots, n)$ .