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Positivity of permutation pattern character polynomials

Christian Gaetz, Laura Pierson

Abstract

Let $N_σ(π)$ denote the number of occurrences of a permutation pattern $σ\in S_k$ in a permutation $π\in S_n$. Gaetz and Ryba (2021) showed using partition algebras that the $d$-th moment $M_{σ,d,n}(π)$ of $N_σ$ on the conjugacy class of $π$ is given by a polynomial in $n,m_1,\dots,m_{dk}$, where $m_i$ denotes the number of $i$-cycles of $π$. They also showed that the coefficient $\langle χ^{λ[n]}, M_{σ,d,n}\rangle$ agrees with a polynomial $a_{σ,d}^λ(n)$ in $n$. This work is motivated by the conjecture that when $σ=\text{id}_k$ is the identity permutation, all of these coefficients are nonnegative. We directly compute closed forms for the polynomials $a_{\text{id}_k}^λ(n)$ in the cases $λ=(1),(1,1),$ and $(2)$, and use this to verify the positivity conjecture for those cases by showing that the polynomials are real-rooted with all roots less than $k$. We also study the case $a_σ^{(1)}(n)$, for which we give a formula for the polynomials and their leading coefficients.

Positivity of permutation pattern character polynomials

Abstract

Let denote the number of occurrences of a permutation pattern in a permutation . Gaetz and Ryba (2021) showed using partition algebras that the -th moment of on the conjugacy class of is given by a polynomial in , where denotes the number of -cycles of . They also showed that the coefficient agrees with a polynomial in . This work is motivated by the conjecture that when is the identity permutation, all of these coefficients are nonnegative. We directly compute closed forms for the polynomials in the cases and , and use this to verify the positivity conjecture for those cases by showing that the polynomials are real-rooted with all roots less than . We also study the case , for which we give a formula for the polynomials and their leading coefficients.
Paper Structure (17 sections, 17 theorems, 168 equations)

This paper contains 17 sections, 17 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Permutation pattern polynomials
  3. A positivity conjecture
  4. Outline
  5. The cases $a_{\textnormal{id}_k}^\emptyset(n), a_{\textnormal{id}_k}^{(1)}(n),$ $a_{\textnormal{id}_k}^{(2)}(n)$, and $a_{\textnormal{id}_k}^{(1,1)}(n)$
  6. Preliminary lemmas
  7. Closed forms for the polynomials
  8. Proof of the positivity conjecture for $\lambda=\emptyset,(1),(2)$, and $(1,1)$
  9. The case $a_\sigma^{(1)}(n)$
  10. Future directions
  11. Roots of the polynomials
  12. Possible approaches for a combinatorial positivity proof
  13. Positivity for $\sigma \ne \textnormal{id}_k$
  14. Polynomiality proofs
  15. Polynomiality of $M_{\sigma_1,\dots,\sigma_d,n}$
  16. ...and 2 more sections

Key Result

Theorem 1.2

Given any permutation patterns $\sigma_1,\dots,\sigma_d$ (not necessarily distinct and not necessarily the same size) with $\sigma_i\in S_{k_i},$$M_{\sigma_1,\dots,\sigma_d,n}$ is a polynomial in the variables $n,m_1,\dots,m_{k_1+\dots+k_d}$ of degree at most $k_1+\dots+k_d$, where $n$ has degree 1

Theorems & Definitions (64)

  • Definition 1.1
  • Theorem 1.2: cf. Gaetz-Ryba GaRy21, Theorem 1.1(a)
  • Theorem 1.3: cf. Gaetz-Ryba GaRy21, Theorem 1.1(b)
  • Conjecture 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 2.1: see Macdonald Macdonald
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 54 more