Abstract
We study the asymptotic behaviour of the statistic $(\operatorname{des}+\operatorname{ides})_W$ which assigns to an element $w$ of a finite Coxeter group $W$ the number of descents of $w$ plus the number of descents of $w^{-1}$. Our main result is a central limit theorem for the probability distributions associated to this statistic. This answers a question of Kahle-Stump and builds upon work of Chatterjee-Diaconis, Özdemir and Röttger.
Highlights
Statistical and probabilistic methods in the investigation of combinatorial and algebraic objects are powerful tools and reveal deeply rooted connections between those fields
Of greatest significance in probabilistic asymptotics is the central limit theorem (CLT), that is the convergence in distribution of a sequence of random variables, normalised by its mean and its standard deviation, towards the standard Gaussian
This paper’s main result is an equivalent formulation of the central limit theorem for a sequence of random variables that arises from a statistic on sequences of finite Coxeter groups
Summary
Statistical and probabilistic methods in the investigation of combinatorial and algebraic objects are powerful tools and reveal deeply rooted connections between those fields. In the appendix we present a discussion of a geometric perspective on the statistic t in the context of the two-sided analogue of the Coxeter complex defined in [9], as well as a table of moments of the statistics des and t for Coxeter groups of type A and B
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