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 $q$).



Does $q$-Catalan number count subspaces?
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. 


