Revisiting Pattern Avoidance and Quasisymmetric Functions

Abstract

Let $${{\mathfrak {S}}}_n$$ be the nth symmetric group. Given a set of permutations $$\Pi $$, we denote by $${{\mathfrak {S}}}_n(\Pi )$$ the set of permutations in $${{\mathfrak {S}}}_n$$ which avoid $$\Pi $$ in the sense of pattern avoidance. Consider the generating function $$Q_n(\Pi )=\sum _\sigma F_{{{\,\mathrm{Des}\,}}\sigma }$$ where the sum is over all $$\sigma \in {{\mathfrak {S}}}_n(\Pi )$$ and $$F_{{{\,\mathrm{Des}\,}}\sigma }$$ is the fundamental quasisymmetric function corresponding to the descent set of $$\sigma $$. 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 $$n\ge 0$$. The purpose of this paper is to continue their investigation by 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.