  Let S_n be the nth symmetric group. Given a set of permutations Pi we denote
by S_n(Pi) the set of permutations in S_n which avoid Pi in the sense of
pattern avoidance. Consider the generating function Q_n(Pi) = sum_pi F_{Des pi}
where the sum is over all pi in S_n(Pi) and F_{Des pi} is the fundamental
quasisymmetric function corresponding to the descent set of pi. Hamaker,
Pawlowski, and Sagan introduced Q_n(Pi) and studied its properties, in
particular, finding criteria for when this quasisymmetric function is symmetric
or even Schur nonnegative for all n >= 0. The purpose of this paper is to
continue their investigation answering some of their questions, proving one of
their conjectures, as well as considering other natural questions about
Q_n(Pi). In particular we look at Pi of small cardinality, superstandard hooks,
partial shuffles, Knuth classes, and a stability property.
