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We give an easy diagrammatical description of the parabolic Kazhdan-Lusztigpolynomials for the Weyl group $W_n$ of type $D_n$ with parabolic subgroup oftype $A_n$ and consequently an explicit counting formula for the dimension ofthe morphism spaces between indecomposable projective objects in thecorresponding category $\O_0^\p$. As a byproduct we categorify irreducible$W_n$-modules corresponding to pairs of one-line partitions. Finally weindicate the motivation for introducing the combinatorics by connections toSpringer theory, the category of perverse sheaves on isotropic Grassmanniansand to Brauer algebras which will be treated in \cite{ES}.

A graphical description of $(D_n,A_{n-1})$ Kazhdan-Lusztig polynomials