Difference between revisions of "Symmetric sum"
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| − | + | The '''symmetric sum''' <math>\sum_{sym} f(x_1, x_2, x_3, \dots, x_n)</math> of a function <math>f(x_1, x_2, x_3, \dots, x_n)</math> of <math>n</math> variables is defined to be <math>\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>. More generally, a '''symmetric sum''' of <math>n</math> variables is any sum that is unchanged by any [[permutation]] of its variables. More generally still, a '''symmetric function''' of <math>n</math> 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 sum]]s. | |
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== See also== | == See also== | ||
Revision as of 14:37, 17 June 2018
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 any sum that is unchanged by any permutation of its variables. More generally still, a symmetric function of 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.
See also
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