This thread is following the question posted here and seeks to find a generalization to the *test function* used in the previous question i.e. $f_a(x_1, \dotsc,x_n)=(x_1+a)\dotsb(x_n+a)$

At first, I propose this conjecture:

$\forall$ $n>0 \in \mathbb N$, $ \exists$ $p>0 \in \mathbb N$ such that
for any set of integers $(x_1,...,x_n)$ and $1\leq x_i \leq n$:

$(x_1,\dotsc,x_n)$ is a permutation of $(1,\dotsc,n)$ if and only if:

$x_1^p+\dotsb+x_n^p=1^p+\dotsb+n^p$

I checked this conjecture for $n=1,\dotsc,17$

The minimum values of $p$ are: $(1,1,2,2,2,4,5,5,5,5,5,7,7,7,7,7,7)$

Here the test function is $f_p(x_1, \dotsc,x_n)=x_1^p+\dotsb +x_n^p$

The question is then: assuming $f_p(x_1, \dotsc,x_n)$ is a symmetric function of n variables with a parameter $p$, what kind of properties must $f$ have in order to be a *test function* i.e. to be able to say if $(x_1,...,x_n)$ is a permutation of $(1,...,n)$?

New examples ?