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 \(\mathcal {F}_x\) is the subset of x-stable complete flags. In the case \(x^2=0\) , the components of \(\mathcal F_x\) are parameterized by Young tableaux of shape \((2^k,1^{n-2k})\) of two columns, where \(k=\textrm{Rank}\,x\) . In this paper, we present an equivalent parameterization of the components of \(\mathcal F_x\) , 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 \(T\in Tab_{(2^k,1^{n-2k})}\) such that \(\mathcal {F}_T\) is a smooth is given by \(\begin{aligned}\left\{ \begin{array}{ll} 2\left( {\begin{array}{c}k+1\\ 4\end{array}}\right) -\left( {\begin{array}{c}k\\ 3\end{array}}\right) +\left( {\begin{array}{c}k\\ 2\end{array}}\right) +1,& n=2k, \\ \frac{(k+1)(3kn-(k+2)(4k-3))}{6},& n\ge 2k+1, \end{array}\right. \end{aligned}\) for all \(n\ge 2k\ge 2\) .
