Counting two-column Young tableaux corresponding to smooth components of Springer fibers

Let x be any nilpotent endomorphism of a vector space V of finite dimension over an algebraically closed field of arbitrary characteristic. The Springer fiber Fx is the subset of x-stable complete flags. In the case x2=0, the components of Fx are parameterized by Young tableaux of shape (2k,1n-2k) of two columns, where k=Rankx. In this paper, we present an equivalent parameterization of the components of Fx, and then we count the number of Young tableaux T of two columns according to the complexity of T. In particular, we show that the number of Young tableaux T∈Tab(2k,1n-2k) such that FT is a smooth is given by (Formula presented.) for all n≥2k≥2.