I want to know if there is some type of combinatorial formula for computing the Betti numbers of the isotropic grassmannian $IG(r,2n)$ for $r\leq n$.
I'm thinking of this as the homogenous space $G/P$ where $G=Sp(2n)$, and $P$ is the maximal parabolic generated by the subset of simple roots $S-{\alpha_r}$, where $S={\alpha_1,...,\alpha_n}$ are the simple roots for $Sp(2n)$.
I know in the case of the Grassmannian $Gr(r,n)$ the $i^{th}$ Betti number is the number of partitons of the integer $i$ which can for inside a $r\times (n-r)$ box. These are the coefficients of the Poicare series of the Grassmanian which happens to be the Gaussian polynomial or q-binomial coefficient http://mathworld.wolfram.com/q-BinomialCoefficient.html
So I wonder if there is a similar combinatorial description for the betti numbers of the isotropic grassmannian or it the Poincare Series has a known form. I can write down the Poincare series for the isotrpic Grassmannian, but its not clear to me if this has some more well-known form.
In particular I'd like to know whether $dim(H^i(IG,r,2n))$ is still bound by the number of partitions of the integer $i$ ?