Difference between revisions of "Symmetric sum"
(Added in the definition of symmetric function and cleaned things up a bit.) |
m |
||
| Line 1: | Line 1: | ||
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 a sum that is unchanged by any [[permutation]] of its variables. More generally still, a '''symmetric function''' of <math>n</math> variables is a function that is unchanged by any [[permutation]] of its variables. | 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 a sum that is unchanged by any [[permutation]] of its variables. More generally still, 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> satisfies <cmath>\sum_{sym} f(x_1, x_2, x_3, \dots, x_n) = n!f(x_1, x_2, x_3, \dots, x_n).</cmath> | |
Any symmetric sum can be written as a polynomial of [[elementary symmetric sum]]s. | Any symmetric sum can be written as a polynomial of [[elementary symmetric sum]]s. | ||
Revision as of 14:54, 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 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.
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
This article is a stub. Help us out by expanding it.
