  In this paper we reduce the problem of counting the number of connected
components in the intersection of two opposite open Schubert cells in the
variety of real complete flags to a purely combinatorial question of counting
the number of orbits of a certain intriguing group action in the space of upper
triangular matrices with {0,1}-valued entries. The crucial step of our
reduction uses the parametrization of the space of real unipotent totally
positive upper triangular matrices introduced by Lusztig and Berenstein, Fomin,
Zelevinski.
