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arXiv · 1109.4624

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

Thomas Bliem·Stavros Kousidis
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Abstract

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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Thomas Bliem, Stavros Kousidis. 2011-09-21. The number of flags in finite vector spaces: Asymptotic normality and Mahonian statistics. https://doi.org/10.1007/s10801-012-0373-1

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