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Feb 14 at 3:11 comment added Ben Webster @YCor Shoot, you're right. Well, there goes that proof method (I had hoped to use this as an intermediate step to a result which is probably still true).
Feb 13 at 8:39 comment added YCor What about $(0,1,2,3,4,5,6)\mapsto (2,5,0,3,6,1,4)$? It has 9 inversions, 5 with equal parity, 4 with opposite parity. (I understand "inversion" as $(i,j)$ such that $i<j$ and $f(i)>f(j)$ and equal/opposite parity according to whether $j-i$ is even or odd.)
Feb 13 at 2:00 history edited LSpice CC BY-SA 4.0
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Feb 12 at 23:02 history asked Ben Webster CC BY-SA 4.0