  We compute for reflection groups of type $A,B,D,F_4,H_3$ and for dihedral
groups a statistic counting the maximal cardinality of a set of elements in the
group whose generalized inversions yield the full set of inversions and which
are minimal with respect to this property. We also provide lower bounds for the
$E$ types that we conjecture to be the exact value of our statistic.
