q-Catalan numbers from Grassmannians In this question by $q$-Catalan numbers I mean the $q$-analog given by the formula $\frac{1}{[n+1]_q}\left[{2n\atop n}\right]_q$. The polynomial $\left[{2n\atop n}\right]_q$ represents the class of the Grassmannian $G(n,2n)$ in the Grothendieck ring of varieties, and $[n+1]$ represents the class of $\mathbb P^n$. Is there a geometric reason why the fraction $[G(n,2n)]/[\mathbb P^n]$ is a polynomial in $[\mathbb A^1]$?
I guess one could ask more generally about why $\frac{[\mathbb P^r][G(k,2k+r)]}{[\mathbb P^k]}$ is a polynomial.
 A: There are a few nice answers to related questions. Unfortunately none of them quite answers the question you asked.


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*The $q$-Catalan number $\frac{1}{[n+1]_q}{ 2n \brack n}_q$ is the Hilbert series of a fairly natural graded representation of the symmetric group $S_n$ coming from an irreducible representation of a rational Cherednik algebra. This was originally proved by Berest-Etingof-Ginzburg and greatly generalized by Gordon-Griffeth:


http://arxiv.org/abs/0912.1578


*Define the "other" $q$-Catalan number as the sum of $q^{|D|}$ where $D$ ranges over "Dyck paths" from $(0,0)$ to $(n,n)$ staying weakly above the diagonal and where $\binom{n}{2}-|D|$ is the number of unit squares between $D$ and the diagonal. Gorsky-Mazin proved that this other $q$-Catalan number evaluated at $t^2$ is the Poincare series of the "Jacobi factor" of the plane curve singularity $x^n=y^{n+1}$:


http://arxiv.org/abs/1105.1151


*The whole story generalizes to the "rational $(q,t)$-Catalan numbers" $\mathrm{Cat}_{a,b}(q,t)$ where $a,b$ are positive coprime integers. The $q$-Catalan you mentioned comes from the formula $$q^{(a-1)(b-1)/2}\mathrm{Cat}_{a,b}(q,q^{-1})=\frac{1}{[a+b]_q}{ a+b \brack a}_q$$ by setting $(a,b)=(n,n+1)$ and the "other" $q$-Catalan number comes from setting $t=1$. The rational $(q,t)$-Catalan numbers are related to many things including the HOMFLY-PT polynomial of torus knots.


See here for some expositions:
http://www.math.miami.edu/~armstrong/Talks/RCC_AIM.pdf
http://thales.math.uqam.ca/~nwilliams/docs/AIM%202012/RCCAIMOutlineOnline.pdf
http://aimath.org/pastworkshops/rationalcatalanrep.pdf
https://www.math.ucdavis.edu/~egorskiy/Presentations/qtcat.pdf
