q-factorial is generating functions of inversions of permutations

Statement

[n]_{q}! = \sum_{w \in S_{n}} q^{inv (w)}

where $[n]_{q}!$ is the q-factorial, $inv$ is the number of inversion of a permutation.
Note that we may replace $inv$ with any Mahonian statistic.

Proof

Note that we may obtain any permutation, $w$ , on $n$ letters by inserting $n$ into a permutation of length $n - 1$ , $u$ . Then, the inversions for these two are related with

inv (w) = inv (u) + (n - index (n))

In other words, we gain as many inversions as how far we are from the last spot when we add this $n$ . If we add it at the end, it adds 0 inversions. If we add it at the beginning, then it would be an inversion with everything else, so $n - 1$ .

Thus we have the relation

\sum_{w \in S_{n}} q^{inv (w)} = (\sum_{u \in S_{n - 1}} q^{inv (u)}) (1 + q^{1} + q^{2} + \dots + q^{n - 1})

and we note that the latter term in the product is $[n]_{q}$ . Thus, it satisfies the same relation $[n]_{q}! = [n - 1]_{q}! [n]_{q}$ .