Reflection ordering on the group G(m,m,n)

In the paper Shi (2008) [13], we introduced a partial ordering, called the reflection ordering, on the elements of G(m,p,n) and described such an ordering on the groups G(m,1,n), m⩾1. In the present paper, we describe the reflection ordering on the group G(m,m,n). As a by-product, we obtain a formula for the enumeration of a certain subset in the symmetric group Sn, which is of independent combinatorial interest.