What are some important combinatorial and algebraic interpretations of the coefficients in the polynomial

$$[n]!_q = (1+q)(1+q+q^2) \ldots (1+q+\cdots + q^{n-1})?$$

As motivation, I will give three interpretations, ask for a fourth, and raise a related question about the unimodality. I would be particularly interested in answers using the RSK-correspondence or subspaces of $\mathbb{F}_q^n$.

Given a permutation $\sigma \in \mathrm{Sym}_n$, let $\mathrm{inv}(\sigma)$ denote the number of

*inversions*of $\sigma$; that is, pairs $(x,y)$ with $x < y$ and $\sigma(x) > \sigma(y)$. Then $[n]!_q = \sum_{\sigma \in \mathrm{Sym}_n} q^{\mathrm{inv}(\sigma)}$.An element $x \in \{1,\ldots, n-1\}$ is a

*descent*of $\sigma \in \mathrm{Sym}_n$ if $\sigma(x) > \sigma(x+1)$. The*major index*$\mathrm{maj}(\sigma)$ is the sum of the descents of $\sigma$. Then $[n]!_q = \sum_{\sigma \in \mathrm{Sym}_n} q^{\mathrm{maj}(\sigma)}$. I think this is due to MacMahon.In the 'inside-out' version of the Fisher–Yates shuffle on an $n$-card deck, at step $j-1$, card $j-1$ from the top is swapped with one of cards in positions $0, 1, \ldots, j-1$ from the top, chosen uniformly at random. These choices are enumerated by $1 + q + \cdots + q^{j-1}$. After $n$ steps (starting with $j=1$), each permutation has equal probability. (This is essentially coset enumeration in the symmetric group by the chain $\mathrm{Sym}_1 \le \mathrm{Sym}_2 \le \ldots \le \mathrm{Sym}_n$.) Hence $[n]!_q$ enumerates permutations according to the sum of the positions chosen at each stage.

Does the normal Fisher–Yates shuffle have a similar combinatorial interpretation? Is there a more natural interpretation of the $q$-power, still using the inside-out Fisher–Yates shuffle?

Finally, (1) makes it easy to see that $[n]!_q$ is symmetric, i.e. the coefficients of $q^m$ and $q^{\binom{n}{2}-m}$ are the same: use the Coxeter involution, thinking of $[n]_q!$ as the Poincaré series of the Coxeter group $\mathrm{Sym}_n$. This can also be seen in a similar way from (2). But it does not seem to be obvious from (3).

Which interpretation is the best way to show that the coefficients in $[n]!_q$ are unimodal, i.e. first increasing then decreasing?