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 $p_0$?
