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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 <math>f(x_1, x_2, x_3, \dots, x_n)</math> is a function of <math>n</math> variables then the symmetric sum <math>\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)})</math>, where <math>\sigma</math> ranges over all permutations of <math>(1, 2, 3, \dots, n)</math>.

All symmetric sums can be written as a polynomial of [[elementary symmetric sum]]s. 

== See also==
*[[Cyclic sum]]
*[[Muirhead's_Inequality]]

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[[Category:Elementary algebra]]
[[Category:Definition]]