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The Inversions function counts the number of inversions in permutation p.

More info: https://reference.wolfram.com/language/Combinatorica/ref/Inversions.html

When I try it, I didn't got a number as the result.

How to use it correctly?

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    $\begingroup$ In the future, please post your code as text, rather than as images. $\endgroup$ – MarcoB Jun 15 '20 at 19:49
  • $\begingroup$ OK, thank you for the suggestion. I'll follow it. @MarcoB $\endgroup$ – Just a learner Jun 15 '20 at 19:50
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You need to convert your permutation from a cycles form into a positional form, and note that you need to provide the missing 3, which I've put in a separate cycle because it's unchanged by the permutation:

perm = FromCycles[{{1, 4, 2, 5}, {3}}];

ToInversionVector[perm]
(* returns: {4, 3, 2, 0} *)

Inversions[perm]
(* returns: 9 *)
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  • $\begingroup$ Thank you for the answer! Btw, where can I learn more about cycles form and positional form? I searched around and not sure if I'm on the correct documentation page. $\endgroup$ – Just a learner Jun 15 '20 at 19:53
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    $\begingroup$ The Combinatorica documentation isn't too detailed, but it has a ToCycles[p] function, so I guessed that the permutations were like this. Also, if you have a cycles expressed permutation with no other cycles like for example {1,2}, make sure to do FromCycles[{{1,2}}] with a double pair of {} to get the positional form. $\endgroup$ – flinty Jun 15 '20 at 20:01

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