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arXiv · 2510.17116

Pattern-Avoiding Peak Functions

Matthew Slattery-Holmes
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Abstract

In 2020, Hamaker, Pawlowski, and Sagan introduced the \emph{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 \emph{pattern-avoiding peak function} that is symmetric, and give explicit formulas for the positive expansion into the closely-related Schur $Q$-functions.

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Matthew Slattery-Holmes. 2025-10-20. Pattern-Avoiding Peak Functions. https://arxiv.org/abs/2510.17116

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