  Natural q analogues of classical statistics on the symmetric groups $S_n$ are
introduced; parameters like: the q-length, the q-inversion number, the
q-descent number and the q-major index. MacMahon's theorem about the
equi-distribution of the inversion number and the reverse major index is
generalized to all positive integers q. It is also shown that the q-inversion
number and the q-reverse major index are equi-distributed over subsets of
permutations avoiding certain patterns. Natural q analogues of the Bell and the
Stirling numbers are related to these q statistics -- through the counting of
the above pattern-avoiding permutations.
