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On permutation

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

\displaystyle 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.

Solution

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}

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