  We introduce the notion of a weighted inversion statistic on the symmetric
group, and examine its distribution on each conjugacy class. Our work
generalizes the study of several common permutation statistics, including the
number of inversions, the number of descents, the major index, and the number
of excedances. As a consequence, we obtain explicit formulas for the first
moments of several statistics by conjugacy class. We also show that when the
cycle lengths are sufficiently large, the higher moments of arbitrary
permutation statistics are independent of the conjugacy class. Fulman (J. Comb.
Theory Ser. A., 1998) previously established this result for major index and
descents. We obtain these results, in part, by generalizing the techniques of
Fulman (ibid.), and introducing the notion of permutation constraints. For
permutation statistics that can be realized via symmetric constraints, we show
that each moment is a polynomial in the degree of the symmetric group.
