Q-multinomial coefficient combinatorial meaning

Question

Could anyone please explain to me a combinatorial meaning for the expression ${n \choose{k_1,k_2,...,k_r}} _q$,where $_q$ means a q-multinomial coefficient. I understand what a normal multinomial coefficient means, but when I consider the q-analog I am quite confused on what interpretation to give to this expression.

Answer 1

Perhaps someone more familiar with the subject will be able to provide more, but one combinatorial interpretation can be found in Theorem $\mathbf{5.1}$ of this PDF. If $W$ is the set of all rearrangements of the word $1^{k_1}2^{k_2}\ldots r^{k_r}$, where $k_1+\ldots+k_r=n$, then

$$\binom{n}{k_1,\ldots,k_r}_q=\sum_{w\in W}q^{\operatorname{inv}w}\;,$$

where $\operatorname{inv}w$ is the number of inversions of $w$. The authors note that some of the common combinatorial interpretations of the $q$-binomial coefficient don’t extend easily to the $q$-multinomial.

Answer 2

The multinomial coefficient $\binom{n}{k_1,\dotsc,k_r}_q$ counts the number of flags $$ 0=V_0\subset \dotsb \subset V_r=\mathbf F_q^n $$ of linear subspaces such that $\dim V_i/V_{i-1} = k_i$ for $i=1,\dotsc,r$. See, for example, Section 5 of Counting subspaces of a finite vector space.