# Relation between groups $A_n$, $B_n$, $D_n$ and $S_n$ or inversions of random elements in Coxeter groups

First of of all I'm trying to find a general interpretation to the following facts below.

1. 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 $$M(n+1)/M(n)=n-\frac 12+o(1)$$

2. 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.

3. 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 highly welcomed.

• The proof of Thm 3.1 and the computations in Cor. 3.2 are simple finger exercises, see Prop. 3.3 and its proof. At the moment, I don't have time to look at your question in detail, but I wonder if you could formulate your question to find properties on general sequences in Prop. 3.3 to obtain such a "result in the similar 'structure'". – Christian Stump Apr 14 at 11:29
• @ Christian Stump Thank you. Sorry, I understand your way: to formulate the question based on Prop. 3.3 and find the properties on its general sequences. However, I am not sure how to do it. – Mikhail Gaichenkov Apr 14 at 17:00