An inversion statistic on the generalized symmetric groups

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 Bn(m). We also establish one-to-one correspondence between all positive integers in the set {1,⋯,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.