Difference between revisions of "Symmetric sum"

Revision as of 16:48, 17 June 2018

The symmetric sum $\sum_{sym} f(x_1, x_2, x_3, \dots, x_n)$ of a function $f(x_1, x_2, x_3, \dots, x_n)$ of $n$ variables is defined to be $\sum_{\sigma} f(x_{\sigma(1)}, x_{\sigma(2)}, x_{\sigma(3)}, \dots, x_{\sigma(n)})$ , where $\sigma$ ranges over all permutations of $(1, 2, 3, \dots, n)$ .

More generally, a symmetric sum of $n$ 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 $n$ variables is a function that is unchanged by any permutation of its variables. The symmetric sum of a symmetric function $f(x_1, x_2, x_3, \dots, x_n)$ therefore satisfies $\sum_{sym} f(x_1, x_2, x_3, \dots, x_n) = n!f(x_1, x_2, x_3, \dots, x_n).$

Given $n$ variables $x_1,\ldots,x_n$ and a symmetric function $f(x_1,\ldots,x_r)$ with $r\leq n$ , the notation $\sum_{sym}f(x_1, x_2, x_3, \dots, x_r)$ is also sometimes used to denote the sum of $f(x_1,\ldots,x_r)$ over all $\left(\begin{matrix}n\cr r\end{matrix}\right)$ subsets of size $r$ in $\{x_1,\ldots,x_n\}$ .

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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 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>.
-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>
+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.
+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_{sym} f(x_1, x_2, x_3, \dots, x_n) = n!f(x_1, x_2, x_3, \dots, x_n).</cmath>
+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_{sym}f(x_1, x_2, x_3, \dots, x_r)</math> is also 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>.
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Difference between revisions of "Symmetric sum"

Revision as of 16:48, 17 June 2018

See also