According to Richard Stanley's answer from normal approximation to $inv(\pi)$: $$ \left| P\left( \frac{\mathrm{inv}(\pi)-\frac 12{n\choose 2}}{\sqrt{n(n-1)(2n+5)/72}}\leq x\right)-\Phi(x)\right| \leq \frac{C}{\sqrt{n}}, $$ where $\Phi(x)$ denotes the standard normal distribution we get that Raw maxima $M(n)$ of Mahonian numbers ($T(n,k)$ is the number of permutations of ${1..n}$ with $k$ inversions): $M(n+1)/M(n)=n-\frac 12+o(1)$ for large $n$.

To be more precise the asymptotic is like $M(n+1)/M(n)=n-\frac {1}{2}+O(\frac {1}{n^{1-\epsilon}})$

I wonder how to modify the normal distribution (for eg. its mean and variance) to take into account the precise information about the $M(n)$ numbers? What will be the new variance, mean? I suppose that we'll have the same mean but a different variance in normal distribution. I understand we cannot improve the inequality without further assumptions about the numbers. Thank you for explanations and ideas.


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