  We show that if a permutation statistic can be written as a linear
combination of bivincular patterns, then its moments can be expressed as a
linear combination of factorials with constant coefficients. This generalizes a
result of Zeilberger. We use an approach of Chern, Diaconis, Kane and Rhoades,
previously applied on set partitions and matchings. In addition, we give a new
proof of the central limit theorem (CLT) for the number of occurrences of
classical patterns, which uses a lemma of Burstein and Hasto. We give a simple
interpretation of this lemma and an analogous lemma that would imply the CLT
for the number of occurrences of any vincular pattern. Furthermore, we obtain
explicit formulas for the moments of the descents and the minimal descents
statistics. The latter is used to give a new direct proof of the fact that we
do not necessarily have asymptotic normality of the number of pattern
occurrences in the case of bivincular patterns. Closed forms for some of the
higher moments of several popular statistics on permutations are also obtained.
