and each symbol is at most $n$...

Any ideas about where to find previous work?

I can find the answers for small $n$ using a brute force search, but I'd like to know more generally.

For example, $1^3+2^3+3^3+4^3+5^3+6^3+7^3 = 784$

Of the sequences of length $n$ using the symbols in $\{1,\ldots, n\}$, 5460 of them have this same sum of cubes. $5460-7!$ of them are not permutations of $[n]$. The same thing happens with the sum of fourth powers (but not when $p=5$).

For which $p$ is the norm of a permutation of $[n]$ distinct from the norm of these other sequences?

Otherwise, what is the proportion of "false positives"?

Is there a $p_0$ for which $p>p_0$ always gives this distinction? What is $p_0$?