# Revisiting pattern avoidance and quasisymmetric functions

## TL;DR

This paper explores the properties of generating functions associated with pattern-avoiding permutations, addressing symmetry, Schur nonnegativity, and other structural questions, building on prior work and proving a conjecture.

## Contribution

It advances understanding of quasisymmetric functions for pattern-avoiding permutations by proving a conjecture and analyzing various classes of permutation sets.

## Key findings

- Proved one of Hamaker, Pawlowski, and Sagan's conjectures.
- Identified conditions for symmetry and Schur nonnegativity of Q_n(Pi).
- Analyzed specific permutation classes like superstandard hooks and Knuth classes.

## Abstract

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.