Consider the $[n]!\_q = \prod\limits\_{k = 1}^{n} \frac{q^k - 1}{q - 1} = \sum\limits\_{k = 0}^{\binom n 2} c\_k q^k$ and let $\{f\_n\}\_{n \in \mathbb{N}}$ be the sequence of the functions on $[0; 1]$ defined by the following $$f\_n(x) = \frac{c\_{\lfloor \binom n 2 x \rfloor}}{n!}$$
Is there a formula for $\lim\limits\_{n \rightarrow \infty} f\_n(x)$? Roughly speaking, what is the limit distribution of its coefficients?



$q$-factorial coefficient asymptotics
Let $Z\_n$ be the number of inversions of a random permutation in $S\_n$. Then for all $x\in\mathbb{R}$, 
$$ \mathrm{Prob}\left(Z\_n<\frac 14 n^2+\frac 16xn^{3/2}\right)\to \mathcal{N}(x),
$$ the standard normal distribution. This goes back to Feller,
1945. See for instance Theorem 3.3.4 of
<https://www.routledgehandbooks.com/doi/10.1201/b18255-6>.


