Counting inversions and descents of random elements in finite Coxeter groups
Abstract
We investigate Mahonian and Eulerian probability distributions given by inversions and descents in general finite Coxeter groups. We provide uniform formulas for the means and variances in terms of Coxeter group data in both cases. We also provide uniform formulas for the double-Eulerian probability distribution of the sum of descents and inverse descents. We finally establish necessary and sufficient conditions for general sequences of Coxeter groups of increasing rank under which Mahonian and Eulerian probability distributions satisfy central and local limit theorems.
Keywords
Cite
@article{arxiv.1802.01389,
title = {Counting inversions and descents of random elements in finite Coxeter groups},
author = {Thomas Kahle and Christian Stump},
journal= {arXiv preprint arXiv:1802.01389},
year = {2019}
}
Comments
26 pages incl. appendix. v2: added further computational data, Remark 6.6 about mod-Gaussian convergence. v3: uniform proof of Theorem 4.1 and a uniform double coset count in Corollary 5.4, completely revised Section 6, v4: final version with corrected Proposition 6.4