B-Orbits in Solutions to the Equation X2 = 0 in Triangular Matrices

Consider In(K), the set of solutions to the equation X2 = 0 in n × n strictly upper-triangular matrices over field K. In this paper we construct the bijection from the set of adjoint orbits in In(K) onto the set of involutions in symmetric group Sn. For a finite field K we compute the number of points in the orbits.