What distribution does the number of inversions of an array of n distinct numbers follow

Asked Aug 15 '20 at 07:35

Active Aug 16 '20 at 00:25

Viewed 56 times

An array of n distinct numbers. I want to know what distribution its number of inversions follows.

I used Mathematica to simulate and initially felt that it obeyed normal distribution:

<< Combinatorica`
data = Table[Inversions@RandomSample[Range[10], 10], 10000];
ListPlot[BinCounts[data, {0, 50, 1}]]

But it's not a strict mathematical proof. I want to know a strict mathematical proof of its distribution.

Here is a similar post. The reference book used in this post is Combinatorics of permutations by Miklos Bona.

ListPlot /@ 
 Table[CoefficientList[Series[QFactorial[n, q], {q, 0, n (n - 1)/2}], 
   q], {n, 9}] (*http://oeis.org/A008302*)

edited Aug 16 '20 at 00:25

asked Aug 15 '20 at 07:35

1

The GF is $$\prod_{k=1}^n\frac{X^k-1}{X-1}$$ – Angina Seng Aug 15 '20 at 07:50
@AnginaSeng Thank you very much. Can you write down the answer in detail? – Please correct GrammarMistakes Aug 16 '20 at 00:05
See http://oeis.org/A008302 for tabulation, discussion, and many links to the literature – Gerry Myerson Aug 16 '20 at 02:17
1

Have you followed up on my suggestion, Please? – Gerry Myerson Aug 17 '20 at 13:09
@GerryMyerson Thank you very much for your guidance. I am reading the content of this post. – Please correct GrammarMistakes Aug 18 '20 at 00:50

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