When $q$ is a prime power, then on the one hand the $q$-binomial coefficient $\binom{n}{k}_q$ equals the number of $k$-dimensional subspaces of $\mathbb{F}_q^n$, and on the other hand it is the generating function of the sequence which sends $r$ to the number of words in two letters $X,Y$ of length $n$ with $k$ occurances of $X$ and with $r$ inversions (i.e. places where $Y$ comes before $X$). Therefore, there must a bijection
$$\{k\text{-dimensional subspaces of } \mathbb{F}_q^n\} \\ \downarrow{\small\cong}\\ \coprod_{r=0}^{k(n-k)} q^r \cdot \{\text{ words in X,Y of length } n \text{ with } k \text{ X's and } r \text{ inversions}\}$$
My question is if you can write down a nice, explicit bijection. I require that its description is independent of the theory of $q$-binomial coefficients, thus doesn't use the calculation of the cardinalities, and that it is also without recursion (since you can easily transform the inductive proof of the equality of cardinalities into a recursive bijection; this doesn't count here).
For $k=1$ the bijection looks as follows: It sends $\langle a_1,\dotsc,a_n \rangle$ to $(a_1/a_{r+1},\dotsc,a_r/a_{r+1}) \cdot Y^r X Y^{n-r-1}$, where $r$ is maximal with $a_{r+1} \neq 0$.