  In this exposition, we compute the hilbert series of the coordinate ring of
some highest weight varieties. We also explain why Narayana numbers (and their
generalizations) appear naturally in the numerator of the hilbert series of the
coordinate ring of complex grassmanian Gr(d,n+d+1) and of the minimal nilpotent
adjoint orbit in sl_n(C). These calculations might be related to some cluster
algebra structures.
