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 (m). We also establish one-to-one correspondence between all positive integers in the set { 1 , ⋯ , 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. [ABSTRACT FROM AUTHOR]