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For any permutation $\sigma:\{1,2,\dots,n\}\to\{1,2,\dots,n\}$ define its displacement as

$$D(\sigma)=\prod_{i=1}^n |i-\sigma(i)|$$

What is greater: the sum of displacements of even permutations or the sum of displacements of odd permutations? The answer may depend on $n$.

The sum of $D(\sigma)$ over the even permutations minus the one over the odd permutations is the determinant of the matrix $A$ with entries $a_{i,j}=\vert i-j\vert$ and this determinant is known to be

$$\det A = (-1)^{n-1} (n-1) 2^{n-2}$$