An inversion statistic on the generalized symmetric groups

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 $B_n^{\left(m\right)}$. We also establish one-to-one correspondence between all positive integers in the set $\{1,\cdots ,m^{n}n!\}$ 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.