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 \(n \geq 0\). 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.
Published: Annals of Combinatorics (2020)