I had completed a paper describing the $q$-Catalan numbers, which is the $q$-analog of the Catalan numbers.


The $n$-th Catalan numbers can be represented by:


$$C\_n=\frac{1}{n+1}{2n \choose n}$$


and with the recurrence relation:


$$C\_{n+1}=\sum^n\_{i=0}C\_i C\_{n-i}\ \ \ \ \ \forall n\geq 0$$


Now, for the $q$-analog, I know the definition of that can be defined as:


$$\lim\_{q\to 1}\frac{1-q^n}{1-q}=n$$


and we know that the definition of the $q$-analog, can be defined like this: 


$$[n]\_q=\frac{1-q^n}{1-q}=1+q+q^2+q^3+\cdots+q^{n-1}$$


which this is the $q$-analog of $n$.


and that for that $q$-analog of ${2n\choose n}$:


$$C\_n(q)=\frac{1}{[n+1]\_q}\begin{bmatrix}2n\\ n\end{bmatrix}\_q$$


**So, everything up to this point I know what I'm doing, and I'm not sure if I did everything correct after this**


So, in order to generate the $q$-Catalan Numbers, I will need to use the Lagrange inversion formula.


And, then I got something like this:


$$G(X)=\sum^\infty\_{i=0}C\_i x^i$$


where $G(x)$ is the generating function, and that


$$G(x)=G\_q(x)=\sum^\infty\_{i=0}C\_n(q)x^n=\sum^\infty\_{i=0}C\_nx^n=1+x+x^2(1+q)+\cdots$$


Since I know that for Catalan Numbers, it's true:


$$G(x)=(G(x))^2+1$$


So, the $q$-analog will just be:


$$G\_q(x)=G(x)G\_q(x)+1$$


So the recurrence relation for the $q$-analog Catalan Numbers:


$$C\_{n+1}(q)=\sum^n\_{i=0}C\_i C\_{n-1}q^i$$


**It just doesn't sound right here...**


Also, I don't have a clue that what does the $q$-Catalan Numbers count, can anyone help me with that or give me like a clue?


Help appreciated!



What does the $q$-Catalan Numbers count?
As Vasu commented already: there is not "the" q-analogue of the Catalan numbers. And indeed, you're mixing two different here.


* Your first q-Catalan numbers defined by the $q$-binomials is MacMahon's q-Catalan numbers which is (and I don't actually know many others) the **major index generating function** on Dyck paths, where the descent set is given by the positions of the valleys.
* Your second $q$-Catalan numbers given by the recurrence is, on the other hand, the **area generating function** on Dyck paths.


Both are deeply related in the context of the $q,t$-Catalan numbers appearing in the theory of symmetric function as a bigraded Hilber series of (the alternating part of) the **space of diagonal coinvariants**.


As Darij mentions, both (and as well the $q,t$-Catalan numbers and the space of diagonal coinvariants) can be found e.g. in Jim Haglund's book [http://www.math.upenn.edu/~jhaglund/books/qtcat.pdf](http://www.math.upenn.edu/%7Ejhaglund/books/qtcat.pdf). You actually find quite a bit as well in our online project <http://www.findstat.org/StatisticsDatabase/St000012> and <http://www.findstat.org/StatisticsDatabase/St000027>.


