Symmetric sum
Revision as of 14:53, 17 June 2018 by Mathematrucker (talk | contribs) (Added in the definition of symmetric function and cleaned things up a bit.)
The symmetric sum 
of a function 
of 
variables is defined to be 
, where 
ranges over all permutations of 
. More generally, a symmetric sum of variables is a sum that is unchanged by any permutation of its variables. More generally still, a symmetric function of variables is a function that is unchanged by any permutation of its variables.
Thus, the symmetric sum of a symmetric function satisfies ![\[\sum_{sym} f(x_1, x_2, x_3, \dots, x_n) = n!f(x_1, x_2, x_3, \dots, x_n).\]](http://latex.artofproblemsolving.com/d/8/2/d8259e8a60509f29bc67fe0ffba11be869dc6f55.png)
Any symmetric sum can be written as a polynomial of elementary symmetric sums.
See also
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