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

TL;DR

This paper investigates the distribution of flags in finite vector spaces, showing they are asymptotically Gaussian, and connects these combinatorial structures to Mahonian statistics and other algebraic objects.

Contribution

It establishes the asymptotic normality of generalized Galois numbers and links them to Mahonian inversion statistics, providing new insights into their combinatorial and algebraic properties.

Findings

Coefficients are asymptotically Gaussian distributed as N grows large.

Generalized Galois numbers relate to Mahonian inversion statistics.

Applications to q-ary codes and symmetric group characters.

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.