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.

  • $\begingroup$ 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'". $\endgroup$ Apr 14 '19 at 11:29
  • $\begingroup$ @ 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. $\endgroup$ Apr 14 '19 at 17:00

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