Difference between revisions of "Symmetric sum"
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| − | A '''symmetric sum''' is any sum in which any [[permutation]] of the variables leaves the sum unchanged. | + | A '''symmetric sum''' is any sum in which any [[permutation]] of the variables leaves the sum unchanged. |
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| + | 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. | All symmetric sums can be written as a polynomial of [[elementary symmetric sum]]s. | ||
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== See also== | == See also== | ||
*[[Cyclic sum]] | *[[Cyclic sum]] | ||
| + | *[[Muirhead's_Inequality]] | ||
{{stub}} | {{stub}} | ||
Revision as of 07:48, 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 
is a function of 
variables then the symmetric sum 
, where 
ranges over all permutations of 
.
All symmetric sums can be written as a polynomial of elementary symmetric sums.
See also
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