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Counting inversions and descents of random elements in finite Coxeter groups

Authors: Thomas Kahle and Christian Stump
Journal: Math. Comp. 89 (2020), 437-464
MSC (2010): Primary 20F55; Secondary 05A15, 05A16, 60F05
Published electronically: May 9, 2019
MathSciNet review: 4011551
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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.

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Additional Information

Thomas Kahle
Affiliation: Fakultät für Mathematik, Otto von Guericke Universität Magdeburg, Magdeburg, Germany
MR Author ID: 869155
ORCID: 0000-0003-3451-5021

Christian Stump
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, Bochum, Germany
MR Author ID: 904921
ORCID: 0000-0002-9271-8436

Keywords: Permutation statistic, central limit theorem, Mahonian numbers, Eulerian numbers, Coxeter group
Received by editor(s): April 27, 2018
Received by editor(s) in revised form: November 10, 2018, and February 25, 2019
Published electronically: May 9, 2019
Additional Notes: The first author acknowledges support from the DFG (314838170, GRK 2297 MathCoRe).
The second author was supported by the DFG grants STU 563/2 “Coxeter-Catalan combinatorics” and STU 563/4-1 “Noncrossing phenomena in Algebra and Geometry”.
Article copyright: © Copyright 2019 American Mathematical Society