# $q$-factorial coefficient asymptotics

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?

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.
Choose $$\xi_i\in \{0,1,\dots,i-1\}$$ uniformly at random. Then $$c_k/n!$$ is a probability that $$\sum \xi_i=k$$. The law of large numbers and central limit theorem work nicely for the distribution of $$\sum \xi_i$$. The limits you are talking about exist and are equal to zero.