On the cohomology of real Grassmann manifolds

We give an explicit and simple construction of the incidence graph for the integral cohomology of real Grassmann manifold Gr(k,n) in terms of the Young diagrams filled with the letter q in checkered pattern. It turns out that there are two types of graphs, one for the trivial coefficients and other for the twisted coefficients, and they compute the homology groups of the orientable and non-orientable cases of Gr(k,n) via the Poincar\'e-Verdier duality. We also give an explicit formula of the Poincar\'e polynomial for Gr(k,n) and show that the Poincar\'e polynomial is also related to the number of points on Gr(k,n) over a finite field {F}_q with q being a power of prime which is also used in the Young diagrams.