The cohomology of $\text{Gr}_2(\mathbb{R}^{4,1})$
Poincaré polynomial: $$x^{4} y^{2} + x^{3} y + 2 x^{2} y + x y + 1$$ Generator count grid:
1
1
2
2
1
1
Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_2(\mathbb{R}^{4,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2.$$