2025/10/20 by Matthew Slattery-Holmes, Slattery-Holmes, Matthew
Materials Science · Mathematics · #05A05 #05E05 #05E10 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2510.17116
openalex publication_date 2025/10/20 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28
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 \mathfrakS3 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 \mathfrakS3 give a pattern-avoiding peak function that is symmetric, and give explicit formulas for the positive expansion into the closely-related Schur Q-functions.