An inversion statistic on the generalized symmetric groups
ADVANCES IN APPLIED MATHEMATICS, vol.154, pp.1-20, 2024 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 154
- Publication Date: 2024
- Doi Number: 10.1016/j.aam.2023.102655
- Journal Name: ADVANCES IN APPLIED MATHEMATICS
- Journal Indexes: Scopus, Aerospace Database, Science Citation Index Expanded (SCI-EXPANDED), Academic Search Premier, Communication Abstracts, Compendex, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
- Page Numbers: pp.1-20
- Keywords: Complex reflection group, Root system, Permutation statistic, Inversion number, Major index
- Erciyes University Affiliated: Yes
Abstract
In this paper, we construct a mixed-base number system over the generalized symmetric group G(m, 1, n), which is a complex reflection group with a root system of type B(m) n . We also establish one-to-one correspondence between all positive integers in the set {1, center dot center dot center dot, mnn!} and the elements of G(m, 1, n) by constructing the subexceedant function in relation to this group. In addition, we provide a new enumeration system for G(m, 1, n) by defining the inversion statistic on G(m, 1, n). Finally, we prove that the flag-major index is equi-distributed with this inversion statistic on G(m, 1, n). Therefore, the flag-major index is a Mahonian statistic on G(m, 1, n) with respect to the length function L. (c) 2023 Elsevier Inc. All rights reserved.