Difference between revisions of "Symmetric sum"

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== See also==
 
== See also==
 
*[[Cyclic sum]]
 
*[[Cyclic sum]]
*[[Muirhead's_Inequality]]
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*[[Muirhead's Inequality]]
  
 
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Revision as of 08:49, 21 October 2017

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 $(1, 2, 3, \dots, n)$.

All symmetric sums can be written as a polynomial of elementary symmetric sums.

See also

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