  We prove several general formulas for the distributions of various
permutation statistics over any set of permutations whose quasisymmetric
generating function is a symmetric function. Our formulas involve certain kinds
of plethystic substitutions on quasisymmetric generating functions, and the
permutation statistics we consider include the descent number, peak number,
left peak number, and the number of up-down runs. We apply these results to
cyclic permutations, involutions, and derangements, and more generally, to
derive formulas for counting all permutations by the above statistics jointly
with the number of fixed points and jointly with cycle type. A number of known
formulas are recovered as special cases of our results, including formulas of
D\'esarm\'enien-Foata, Gessel-Reutenauer, Stembridge, Fulman, Petersen,
Diaconis-Fulman-Holmes, Zhuang, and Athanasiadis.
