Poincaré polynomial at -1 associated with a Young diagram of three rows

In this paper, we give an explicit formula for the Poincaré polynomial P λ ( x ) for the Betti numbers of the Springer fibers over nilpotent elements in g l n ( C ) of Jordan form λ = a b c with a ≥ b ≥ c ≥ 0 at x = - 1 . In particular, we introduce λ -vacillating diagrams and show that P ab ( - 1 ) is equal to the number of restricted Dyck paths.