The number of flags in finite vector spaces: asymptotic normality and Mahonian statistics

We study the generalized Galois numbers which count flags of length r in N -dimensional vector spaces over finite fields. We prove that the coefficients of those polynomials are asymptotically Gaussian normally distributed as N becomes large. Furthermore, we interpret the generalized Galois numbers as weighted inversion statistics on the descent classes of the symmetric group on N elements and identify their asymptotic limit as the Mahonian inversion statistic when r approaches ∞. Finally, we apply our statements to derive further statistical aspects of generalized Rogers–Szegő polynomials, reinterpret the asymptotic behavior of linear q -ary codes and characters of the symmetric group acting on subspaces over finite fields, and discuss implications for affine Demazure modules and joint probability generating functions of descent-inversion statistics.