It is known that the coefficient of $q^t$ in Gaussian binomial coefficient $\binom{m+n}m_q$ equals the number of permutations of the multiset $\{0^m, 1^n\}$ with $t$ inversions.
Is there a closed formula for the polynomial $f_{m,n,k}(q_{01},q_{02},q_{12})$, where the coefficient of $q_{01}^{t_{01}} q_{02}^{t_{02}} q_{12}^{t_{12}}$ equals the number of permutations of the multiset $\{0^m, 1^n, 2^k\}$ with $t_{ij}$ inversions formed by elements $i,j\in\{0,1,2\}$?
Clearly, we should have the following specializations: $$\begin{cases} f_{m,n,k}(1,1,1) = \binom{m+n+k}{m,\ n,\ k}, \\ f_{m,n,k}(q,1,1) = \binom{m+n+k}k\binom{m+n}m_{q}, \\ f_{m,n,k}(1,q,1) = \binom{m+n+k}n\binom{m+k}m_{q}, \\ f_{m,n,k}(1,1,q) = \binom{m+n+k}m\binom{n+k}n_{q}, \\ f_{m,n,k}(1,q,q) = \binom{m+n}m\binom{m+n+k}k_{q}, \\ f_{m,n,k}(q,1,q) = \binom{m+k}m\binom{m+n+k}n_{q}, \\ f_{m,n,k}(q,q,1) = \binom{n+k}n\binom{m+n+k}m_{q}, \\ f_{m,n,k}(q,q,q) = \binom{m+n+k}{m,\ n,\ k}_{q}. \end{cases} $$
This question may be relevant.
