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Pattern-Avoiding Peak Functions

Matthew Slattery-Holmes

TL;DR

This paper investigates when pattern-avoiding peak functions $R_n(\Pi)$ are symmetric and Schur $Q$-positive, extending the pattern-avoidance framework of Hamaker–Pawlowski–Sagan to peak statistics. Focusing on $\Pi\subset \mathfrak{S}_3$ with $\{123,321\}\not\subset \Pi$, the authors determine exactly which $\Pi$ yield symmetry for all $n$ and provide explicit $Q_\\lambda$-expansions of $R_n(\Pi)$ for $n\ge 3$, with the results summarized in Table 1. In these symmetric cases, $R_n(\Pi)$ is Schur $Q$-positive and, since $Q_\lambda=2^{\ell(\lambda)}P_\lambda$, this also yields Schur $P$-positivity. The work combines peak-set analysis of standard/shifted tableaux with RS and Sagan-Worley insertions to justify the positive expansions and to relate peak statistics to shifted-tableaux combinatorics.

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.

Pattern-Avoiding Peak Functions

TL;DR

This paper investigates when pattern-avoiding peak functions are symmetric and Schur -positive, extending the pattern-avoidance framework of Hamaker–Pawlowski–Sagan to peak statistics. Focusing on with , the authors determine exactly which yield symmetry for all and provide explicit -expansions of for , with the results summarized in Table 1. In these symmetric cases, is Schur -positive and, since , this also yields Schur -positivity. The work combines peak-set analysis of standard/shifted tableaux with RS and Sagan-Worley insertions to justify the positive expansions and to relate peak statistics to shifted-tableaux combinatorics.

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 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 -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 give a \emph{pattern-avoiding peak function} that is symmetric, and give explicit formulas for the positive expansion into the closely-related Schur -functions.
Paper Structure (2 sections, 1 theorem, 3 equations, 1 table)

This paper contains 2 sections, 1 theorem, 3 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background

Key Result

Theorem 1.1

If $\Pi \subset \mathfrak{S}_3$ and $\{123,321\}\not \subset \Pi$, the pattern-avoiding peak function $R_n(\Pi)$ is symmetric for all $n$ precisely when $\Pi$ is one of the sets given in the left column of Table 1. In all of these cases $R_n(\Pi)$ is Schur $Q$-positive, and for $n\geqslant 3$, the c

Theorems & Definitions (2)

  • Theorem 1.1
  • Example 2.1