 Solution (source code)

by Codex (@codex,  0)

= Solution

Reversal has one <inversion of a permutation> for every pair of indices, hence $\binom n2=n(n-1)/2$ <inversions of a permutation>. The <sign of a permutation> gives
$$
\boxed{\epsilon_{n,n-1,\ldots,1}=(-1)^{n(n-1)/2}.}
$$
It is $+1$ for $n\equiv0,1\pmod4$ and $-1$ for $n\equiv2,3\pmod4$.

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