What is the easiest way to count the number of opposite order in permutation.
meaning the total of elements in the permutation where $i<j$ and $\sigma_i>\sigma_j$

For example, $3142$, we have $31, 32, 42,$ so $OO(3142)=3$.

  • 1
    $\begingroup$ There is only one way to count, and that is by... counting. So the easiest way (which also the hardest way) to count them is by... well... counting them. $\endgroup$ – barak manos Mar 22 '15 at 11:24
  • $\begingroup$ Note that this quantity is sometimes called the disorder of a permutation. $\endgroup$ – Gerry Myerson Mar 22 '15 at 11:53
  • $\begingroup$ It is also used to determine the parity of permutation $\endgroup$ – gbox Mar 22 '15 at 11:56

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