Symmetric functions

In mathematics, a symmetric function of n variables is one whose value at any n-tuple of arguments is the same as its value at any permutation of that n-tuple. So, if e.g. ${\displaystyle f({\mathbf {x}})=f(x_{1},x_{2},x_{3})}$, the function can be symmetric on all its variables, or just on ${\displaystyle (x_{1},x_{2})}$, ${\displaystyle (x_{2},x_{3})}$, or ${\displaystyle (x_{1},x_{3})}$. While this notion can apply to any type of function whose n arguments have the same domain set, it is most often used for polynomial functions, in which case these are the functions given by symmetric polynomials. There is very little systematic theory of symmetric non-polynomial functions of n variables, so this sense is little-used, except as a general definition.

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