I am studying probability model of random permetuation Let $b(n; k)$ denote the number of permutations of {1,...,n} with precisely k inversions ($inv(\pi)$). The analytic approach was considered by L.Clark in article 'An Asymptotic Expansion for the Number of Permutations with a Certain Number of Inversions'. $b(n; k)$ is represented via Bernoulli numbers and Hermite polynomials http://www.combinatorics.org/ojs/index.php/eljc/article/view/v7i1r50/pdf, Research Paper 50, 11 pp. 1

Looking at probabilistic model it is known that

(1) $\left| P\left( \frac{\mathrm{inv}(\pi)-\frac 12{n\choose 2}}{\sqrt{n(n-1)(2n+5)/72}}\leq x\right)-\Phi(x)\right| \leq \frac{C}{\sqrt{n}}$ http://arxiv.org/PS_cache/math/pdf/0508/0508242v2.pdf

My next step was to have more advanced model. So I took into account asymptotic expansions in the central limit theorem (for more details see V. V. Petrov, Sums of independent random variables.):

(2) $F_N (x)=Φ(x)+\frac{3}{50\sqrt{2π}} e^{-x^2/2} (x^3-3x) \frac {6N^3+21N^2+31N+31} {N(2N+5)^2 (N-1)}+O(\frac{1}{N^2} )$ Generally speaking it is possible to calculate the next terms if needed ( also the series convergence - it is possible to prove it).

Actually I am trying to find any explanations about the advantages and disadvantages of the probabilistic model with comparing to the asymptotic expansion via Bernoulli numbers and Hermite polynomials?

Any help is highly welcomed. Thank you in advance.