Does $q$-Catalan number count subspaces? 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 $q$).
 A: 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. 
A: Wikipedia has that 

$q$-binomial $ \binom{n}{k}_q$ counts the subspaces of dimension $k$ in the vector space $\mathbb{F}_q^n$ 

As a non-expert I would hope that with an appropriate group action I could find objects enumerated by 
$$ \frac{q^n}{[n+1]_q} \binom{2n}{n}_q  $$
and I would look through Richard Stanley's Enumerative Combinatorics and hope for the best.

There also seems to be a result from 2010


*

*q,t-Catalan numbers and knot homology
where $q$-Catalan numbers are shown to enumerate invariants related to the Hilbert scheme of $n$ points on $\mathbb{C}^2$ (which respect to an equivariant torus action).
There are clearly some vector spaces related to the $q$-catalan numbers Gorsky has $q,t$ Catalan numbers and just set $t = 1$ ( I am looking for the correct degeneration).  I do not underestand why $(q,t)$-enumeration is so trendy, as one could have arbitrary deformations.
My double-use of the letter $q$ is a bit suspect since $q$ could be:


*

*a prime number

*a unit complex number


and this ambiguity persists in the finite fields / modular forms literature.
