Ordered Bell numbers, Hermite polynomials, skew Young tableaux, and Borel orbits

We give three interpretations of the number b of orbits of the Borel subgroup of upper triangular matrices on the variety X of complete quadrics. First, we show that b is equal to the number of standard Young tableaux on skew-diagrams. Then, we relate b to certain values of a modified Hermite polynomial. Third, we relate b to a certain cell decomposition on X previously studied by De Concini, Springer, and Strickland. Using these, we give asymptotic estimates for b as the dimension of the quadrics increases.