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

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

2011-09-21

arxiv.org/abs/1109.4624v2

Mathematics

Combinatorics

19 pages. Corrected proof of asymptotic normality (Theorem 3.5). Previous Proposition 3.3 is false

Scientific paper

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 infinity. Finally, we apply our statements to derive further statistical aspects of generalized Rogers-Szegoe polynomials, re-interpret 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.

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