# How to modify normal distribution to take into account properties of Mahonian numbers

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.