## proportion of $n$ symbol sequences whose $l^p$ norm is equal to $|[n]|_p$

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 $p=5$).

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

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

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

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 needless to say, I can cut some of the computation by considering multisets of size $n$ on $n$ symbols. – Alejandro Erickson Jun 23 2011 at 15:12