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| − | 1) Starting from left to right, the number of inversions
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| − | for 1st number is n-1
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| − | for 2nd number is n-2
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| − | ...
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| − | ..
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| − | ....nth number is n-n = 0
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| − | Total number of inversions is the sum of all of the above
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| − | sum of integers from 0 to n - 1 = n(n-1)/2
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| − | 2. We know that the maximum num of inversions is n(n-1)/2.
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| − | Consider a permutation P.
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| − | a1 a2 ai..ak...aj..am.... an-1 an
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| − | ai and aj are out of order
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| − | ak and am are in order.
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| − | For Pr,
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| − | ai and aj will get reversed, and will become in order.
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| − | ak and am will get reversed and will become out of order.
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| − | Every In order pair becomes an inversion.
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| − | And every inversion becomes corrected.
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| − | If P contains "x" inversions.. then it also contained "n(n-1)/2 - x" in order pairs.
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| − | Thus Pr contains "x" in order pairs and "n(n-1)/2 - x" inversions.
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| − | Summing up the inversions in P and Pr we get n(n-1)/2
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| − | 3. To calculate the max inversions in part 1,
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| − | every ith position had n - i inversions.
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| − | i.e following a number, all other numbers will be out of order.
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| − | In the average case, we can argue that about 1/2 of the number will be
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| − | out of order....
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| − | for 1st number (n-1)/2
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| − | for 2nd number (n-2)/2
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| − | ...
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| − | ..
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| − | ....nth number is (n-n)/2 = 0
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| − | Total number of inversions is the sum of all of the above
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| − | sum of integers from 0 to (n - 1)/2 = n(n-1)/4...
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| − | '''Alternative solution to 3:''' Any permutation of any set is equally likely. Previous result shows that every permutation has its reversal. If we put pair up every permutation of a set with its reversal, they combine to n(n-1)/2 inversions. There are half as much pairs as there are permutations so the result is n(n-1)/2 * 1/2 = n(n-1)/4
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