A statistic 
on the symmetric group 
is called a weighted inversion statistic if there exists an upper triangular matrix 
such that
 |
where 
is the characteristic function.
The inversion count (
for 
) defined by Cramer (1750) and the major index (
; 
otherwise) defined by MacMahon (1913) are both weighted inversion statistics (Degenhardt and Milne).
