First of of all I’m trying to find a general interperetation to the following facts below.
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Let’s look at the property of Kendall-Mann numbers $ M(n)$ which are row maxima of Triangle of Mahonian numbers $ T(n,k)$ (the number of permutations of {1..n} with k inversions). According to Richard Stanley $ $ \left| P\left( \frac{\mathrm{inv}(\pi)-\frac 12{n\choose 2}}{\sqrt{n(n-1)(2n+5)/72}}\leq x\right)-\Phi(x)\right| \leq \frac{C}{\sqrt{n}}, $ $ where $ \Phi(x)$ denotes the standard normal distribution. From this it is immediate that $ M(n+1)/M(n)=n-\frac 12+o(1)$
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Looking at combinatorial proof for the property of Kendall-Mann numbers numbers at MO $ M(n) \approx c n!/n^{3/2}$ and $ $ \frac{M(n+1)}{M(n)} \approx \frac{(n+1)(n+1)^{-3/2}}{n^{-3/2}} = n (1+1/n)^{-1/2} \approx n-1/2.$ $ This is pretty the same result as #1.
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Reading through Counting inversions and descents of random elements in finite Coxeter groups I noticed Corollary 3.2 (page 6 of the article, pls have a look at it) that the mean and variance of W-Mahonian distribution depend on the types of groups, i.e. $ A_n$ , $ B_n$ , $ D_n$ . By and large it’s about $ n^{3/2}$ like for #1 and #2.
This results in the similar ‘structure’: $ \approx n-1/2$ .
So I wonder why? I am looking for a general explanation to the facts. I guess that it is needed to study relations between groups: $ A_n$ , $ B_n$ , $ D_n$ and $ S_n$ . Any help in explanation of the facts are higly welcomed.
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