Difference between revisions of "Symmetric sum"
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| − | + | The '''symmetric sum''' <math>\sum_{\rm 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 a sum 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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| + | A '''symmetric function''' of <math>n</math> variables is a function that is unchanged by any [[permutation]] of its variables. The symmetric sum of a symmetric function <math>f(x_1, x_2, x_3, \dots, x_n)</math> therefore satisfies <cmath>\sum_{\rm sym} f(x_1, x_2, x_3, \dots, x_n) = n!f(x_1, x_2, x_3, \dots, x_n).</cmath> | ||
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| + | Given <math>n</math> variables <math>x_1,\ldots,x_n</math> and a symmetric function <math>f(x_1,\ldots,x_r)</math> with <math>r\leq n</math>, the notation <math>\sum_{\rm sym}f(x_1, x_2, x_3, \dots, x_r)</math> is sometimes used to denote the sum of <math>f(x_1,\ldots,x_r)</math> over all <math>\left(\begin{matrix}n\cr r\end{matrix}\right)</math> subsets of size <math>r</math> in <math>\{x_1,\ldots,x_n\}</math>. | ||
== See also== | == See also== | ||
*[[Cyclic sum]] | *[[Cyclic sum]] | ||
| − | *[[Muirhead' | + | *[[Muirhead's Inequality]] |
| − | + | *[[PaperMath’s sum]] | |
{{stub}} | {{stub}} | ||
| − | [[Category: | + | [[Category:Algebra]] |
[[Category:Definition]] | [[Category:Definition]] | ||
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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