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$[n]!_q$ gives the number of points in the full flag variety of full flags of subspaces in an n-dimensional vector space $\mathbb{F}_q^n$ over the finite field $\mathbb{F}_q$.

Recall that the full flag variety (over any field) has a natural stratification, the Bruhat stratification. Due to the above point-counting remark, it follows that the coefficient of $q^i$ in $[n]!_q$ is the number of i-dimensional cells in the Bruhat stratification.

There is also a way to deduce the unimodality of the coefficients from this geometric perspective. Namely, the partial order on the Bruhat cells whereby $C \leq C'$ if $C$ is contained in the closure of $C'$ is called the Bruhat order or strong order. The strong order can be viewed as an order on the symmetric group because the Bruhat cells are naturally labeled by permutations. Strong order is graded, and the rank sizes are precisely the coefficients of $[n]!_q$ (i.e., the number of permutations with given inversion number). Richard Stanley showed in the "Weyl groups..." paper cited below that in this situation (when you have a complex projective variety with a cellular decomposition satisfying certain conditions), the poset in question is necessarily graded, rank-symmetric, rank-unimodal, and strongly Sperner, which in particular implies the unimodality of the coefficients of $[n]!_q$. His proof employed the hard Lefschetz theorem and so can hardly be called elementary, but it is conceptual.

Stanley, Richard P., Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods 1, 168-184 (1980). ZBL0502.05004.

This answer concerns a geometric/Lie-theoretic interpretation of $[n]!_q$.

$[n]!_q$ gives the number of points in the full flag variety of full flags of subspaces in an $n$-dimensional vector space $\mathbb{F}_q^n$ over the finite field $\mathbb{F}_q$.

Recall that the full flag variety (over any field) has a natural stratification, the Bruhat stratification. Due to the above point-counting remark, it follows that the coefficient of $q^i$ in $[n]!_q$ is the number of $i$-dimensional cells in the Bruhat stratification.

There is also a way to deduce the unimodality of the coefficients from this geometric perspective. Namely, the partial order on the Bruhat cells whereby $C \leq C'$ if $C$ is contained in the closure of $C'$ is called the Bruhat order or strong order. The strong order can be viewed as an order on the symmetric group because the Bruhat cells are naturally labeled by permutations. Strong order is graded, and the rank sizes are precisely the coefficients of $[n]!_q$ (i.e., the number of permutations with given inversion number). Richard Stanley showed in the "Weyl groups..." paper cited below that in this situation (when you have a complex projective variety with a cellular decomposition satisfying certain conditions), the poset in question is necessarily graded, rank-symmetric, rank-unimodal, and strongly Sperner, which in particular implies the unimodality of the coefficients of $[n]!_q$. His proof employed the hard Lefschetz theorem and so can hardly be called elementary, but it is conceptual.

Stanley, Richard P., Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods 1, 168-184 (1980). ZBL0502.05004.

$[n]!_q$ gives the number of points in the full flag variety of full flags of subspaces in an n-dimensional vector space $\mathbb{F}_q^n$ over the finite field $\mathbb{F}_q$.

Recall that the full flag variety has a natural stratification, the Bruhat stratification. Due to the above point-counting remark, it follows that the coefficient of $q^i$ in $[n]!_q$ is the number of i-dimensional cells in the Bruhat stratification.

$[n]!_q$ gives the number of points in the full flag variety of full flags of subspaces in an n-dimensional vector space $\mathbb{F}_q^n$ over the finite field $\mathbb{F}_q$.

Recall that the full flag variety (over any field) has a natural stratification, the Bruhat stratification. Due to the above point-counting remark, it follows that the coefficient of $q^i$ in $[n]!_q$ is the number of i-dimensional cells in the Bruhat stratification.

There is also a way to deduce the unimodality of the coefficients from this geometric perspective. Namely, the partial order on the Bruhat cells whereby $C \leq C'$ if $C$ is contained in the closure of $C'$ is called the Bruhat order or strong order. The strong order can be viewed as an order on the symmetric group because the Bruhat cells are naturally labeled by permutations. Strong order is graded, and the rank sizes are precisely the coefficients of $[n]!_q$ (i.e., the number of permutations with given inversion number). Richard Stanley showed in the "Weyl groups..." paper cited below that in this situation (when you have a complex projective variety with a cellular decomposition satisfying certain conditions), the poset in question is necessarily graded, rank-symmetric, rank-unimodal, and strongly Sperner, which in particular implies the unimodality of the coefficients of $[n]!_q$. His proof employed the hard Lefschetz theorem and so can hardly be called elementary, but it is conceptual.

Stanley, Richard P., Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods 1, 168-184 (1980). ZBL0502.05004.

It gives the number of points in the full flag variety of full flags of subspaces in an n-dimensional vector space $\mathbb{F}_q^n$ over the finite field $\mathbb{F}_q$.

$[n]!_q$ gives the number of points in the full flag variety of full flags of subspaces in an n-dimensional vector space $\mathbb{F}_q^n$ over the finite field $\mathbb{F}_q$.

Recall that the full flag variety has a natural stratification, the Bruhat stratification. Due to the above point-counting remark, it follows that the coefficient of $q^i$ in $[n]!_q$ is the number of i-dimensional cells in the Bruhat stratification.