Pattern-Avoiding Peak Functions
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909857190248448 |
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| author | Slattery-Holmes, Matthew |
| author_facet | Slattery-Holmes, Matthew |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17116 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pattern-Avoiding Peak Functions Slattery-Holmes, Matthew Combinatorics 05A05, 05E05, 05E10 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. |
| title | Pattern-Avoiding Peak Functions |
| topic | Combinatorics 05A05, 05E05, 05E10 |
| url | https://arxiv.org/abs/2510.17116 |