Symmetric sum

The symmetric sum $\sum_{sym} f(x_1, x_2, x_3, \dots, x_n)$ of a function $f(x_1, x_2, x_3, \dots, x_n)$ of $n$ variables is defined to be $\sum_{\sigma} f(x_{\sigma(1)}, x_{\sigma(2)}, x_{\sigma(3)}, \dots, x_{\sigma(n)})$ , where $\sigma$ ranges over all permutations of $(1, 2, 3, \dots, n)$ . More generally, a symmetric sum of $n$ variables is any sum that is unchanged by any permutation of its variables. More generally still, a symmetric function of $n$ variables is any function that is unchanged by any permutation of its variables.

Any symmetric sum can be written as a polynomial of elementary symmetric sums.

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Symmetric sum

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