Integer Representations of the Generalized Symmetric Groups
Hasan Arslan, Alnour Altoum, Mariam Zaarour

TL;DR
This paper introduces a new mixed-base number system for the generalized symmetric group G(m,1,n), establishing a bijection with integers, defining a new enumeration method, and proving the equidistribution of the flag-major index and inversion statistic.
Contribution
It constructs a novel mixed-base representation and enumeration system for G(m,1,n), linking combinatorial statistics with group elements.
Findings
Established a bijection between integers and group elements.
Defined a new inversion statistic for G(m,1,n).
Proved the flag-major index is Mahonian on G(m,1,n).
Abstract
In this paper, we construct a mixed-base number system over the generalized symmetric group , which is a complex reflection group with a root system of type . We also establish one-to-one correspondence between all positive integers in the set and the elements of by constructing the subexceedant function in relation to this group. In addition, we provide a new enumeration system for by defining the inversion statistic on . Finally, we prove that the \textit{flag-major index} is equi-distributed with this inversion statistic on . Therefore, the flag-major index is Mahonian on with respect to the length function .
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Algebraic structures and combinatorial models · Advanced Mathematical Identities
