Multivariate functions whose value is independent of the order of the arguments Let $r_1, r_2, \ldots, r_k$ be positive integers with or without repetition such that $1\le r_i \le n$ for $i = 1, 2, \ldots, k$. Let $f$ be a continuous multivariate function with the property that the value of $f(r_1, r_2, \ldots, r_k)$ is independent of the order the arguments $r_1, r_2, \ldots, r_k$. The trivial examples are:
$f(r_1, r_2, \ldots, r_k) = g(r_1) + g(r_2) + \ldots + g(r_k)$,
$f(r_1, r_2, \ldots, r_k) = g(r_1)g( r_2) \ldots g(r_k)$ and
$f(r_1, r_2, \ldots, r_k) = c$
where $g$ is a continuous univariate function and $c$ is a constant. I have two questions
Questions:


*

*Are there other non trivial examples of such functions?

*Are there infinitely many such non trivial functions $f$?
 A: You could symmetrize an arbitrary function by averaging it over the symmetric group.
Explicitly: take your favourite function $h$ in $n$ variables, and set $f(r_1, ..., r_n) = \frac{1}{n!}\sum_{\sigma} h(r_{\sigma(1)}, ..., r_{\sigma(n)})$ where the sum is taken over all permutations $\sigma$ of the numbers $1,...,n$.  
I'd love to hear of an example which isn't obtained by symmetrization!  
edit: heh.  Any such example would be its own symmetrization.  So I guess in some sense this is a complete, if overly glib, answer.
A: I don't understand why you say that $f$ is continuous if its inputs are positive integers. Anyway, you can take any symmetric polynomial in the $g(r_i)$, e.g. $\sum_{i < j} g(r_i) g(r_j)$. 
A: Another very famous class of multivariate functions that satisfies this permutation invariance is: Symmetric gauge functions, for example, all the vector $\ell_p$ norms; also in fact their quasi-norm cousins are symmetric.
It is not clear to me where does it matter that the inputs be positive integers.
For example, suppose $r_1,\ldots, r_n$ are $n$ indeterminate elements of some group (possibly non-abelian). Then, using the kind of symmetrization over $S_n$ (that B. Young mentions), we can obtain symmetric functions in an even more general setup.
P.S.: You might also enjoy looking at the book: Symmetric functions and Hall Polynomials by MacDonald.
