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
Are there other non trivial examples of such functions?
Are there infinitely many such non trivial functions $f$?