Pattern-Avoiding Peak Functions

In 2020, Hamaker, Pawlowski, and Sagan introduced the pattern quasisymmetric functions, which are quasisymmetric functions associated with pattern-avoidance classes of permutations, and defined via expansions in fundamental quasisymmetric functions. They determined which subsets of the symmetric group $\mathfrak{S}_3$ index pattern quasisymmetric functions that are symmetric, and showed that these symmetric pattern quasisymmetric functions are also Schur-positive. They then posed the question of when symmetry or Schur $P$-positivity occur for analogous quasisymmetric functions defined in terms of peak functions. In this work we answer this question, that is, we identify precisely which subsets of $\mathfrak{S}_3$ give a pattern-avoiding peak function that is symmetric, and give explicit formulas for the positive expansion into the closely related Schur $Q$-functions.