TITLE: Does $q$-Catalan number count subspaces?
QUESTION [11 upvotes]: Consider the $n$-element subsets $\{a_1<a_2<\cdots <a_n\}$ of $\{1,\ldots ,2n\}$ satisfying $a_i\geq 2i$ for all $i=1,\ldots ,n$. The number of such subsets is given by $${2n\choose n}-{2n\choose n-1}=\frac{1}{n+1}{2n\choose n},$$
which is the $n$th Catalan Number.
I want to know if the $q$-Catalan number $$\frac{q^{n}}{[n+1]_q}{2n\choose n}_q={2n\choose n}_q-{2n\choose n-1}_q$$
counts some kind of special $n$-dimensional subspaces  inside $\mathbb{F}_q^{2n}$? Note that ${2n \choose n}_q$ is the total number of $n$-dimensional subspaces of $\mathbb{F}_q^{2n}$ ($\mathbb{F}_q$ denotes finite field of order $q$).

REPLY [5 votes]: An answer to your question was given in ``Rank Polynomials" by Brandt, Dipper, James, and Lyle, published in Proc. London Math. Soc. (3) 98 (2009), 1-18. A special case of Theorem 2.6 in that paper answers your question.