Difference between revisions of "Symmetric sum"
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*[[Cyclic sum]] | *[[Cyclic sum]] | ||
*[[Muirhead's Inequality]] | *[[Muirhead's Inequality]] | ||
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[[Category:Algebra]] | [[Category:Algebra]] | ||
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Latest revision as of 11:52, 8 October 2023
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.
Any symmetric sum can be written as a polynomial of elementary symmetric sums.
A symmetric function of variables is a function that is unchanged by any permutation of its variables. The symmetric sum of a symmetric function therefore satisfies ![\[\sum_{\rm sym} f(x_1, x_2, x_3, \dots, x_n) = n!f(x_1, x_2, x_3, \dots, x_n).\]](http://latex.artofproblemsolving.com/4/2/1/4219ee21f7bab9f7560a061edddaef1a589b2caf.png)
Given variables 
and a symmetric function 
with 
, the notation 
is sometimes used to denote the sum of over all 
subsets of size 
in 
.
See also
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