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Coefficients and roots of peak polynomials

Sara Billey, Matthew Fahrbach, Alan Talmage

Abstract

Given a permutation $π=π_1π_2\cdots π_n \in \mathfrak{S}_n$, we say an index $i$ is a peak if $π_{i-1} < π_i > π_{i+1}$. Let $P(π)$ denote the set of peaks of $π$. Given any set $S$ of positive integers, define ${\mathcal{P}_S(n)=\{π\in \mathfrak{S}_n:P(π)=S\}}$. Billey-Burdzy-Sagan showed that for all fixed subsets of positive integers $S$ and sufficiently large $n$, $|\mathcal{P}_S(n)|=p_S(n)2^{n-|S|-1}$ for some polynomial $p_S(x)$ depending on $S$. They conjectured that the coefficients of $p_S(x)$ expanded in a binomial coefficient basis centered at $\max(S)$ are all positive. We show that this is a consequence of a stronger conjecture that bounds the modulus of the roots of $p_S(x)$. Furthermore, we give an efficient explicit formula for peak polynomials in the binomial basis centered at $0$, which we use to identify many integer roots of peak polynomials along with certain inequalities and identities.

Coefficients and roots of peak polynomials

Abstract

Given a permutation , we say an index is a peak if . Let denote the set of peaks of . Given any set of positive integers, define . Billey-Burdzy-Sagan showed that for all fixed subsets of positive integers and sufficiently large , for some polynomial depending on . They conjectured that the coefficients of expanded in a binomial coefficient basis centered at are all positive. We show that this is a consequence of a stronger conjecture that bounds the modulus of the roots of . Furthermore, we give an efficient explicit formula for peak polynomials in the binomial basis centered at , which we use to identify many integer roots of peak polynomials along with certain inequalities and identities.

Paper Structure

This paper contains 10 sections, 39 theorems, 63 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. An approach to the positivity conjecture
  3. Roots of peak polynomials
  4. Main results
  5. Peak polynomials with only integral roots
  6. Gap of three independence
  7. Evaluating $p_S(x)$ at nonnegative integers
  8. Connections to alternating permutations
  9. Related work and conjectures
  10. Acknowledgments

Key Result

Theorem \oldthetheorem

If $S$ is a nonempty admissible set and $m=\max(S)$, then for $n\geq m$, where $p_S(x)$ is a polynomial of degree $m-1$ depending on $S$ such that $p_S(n)$ is an integer for all integral inputs $n$. If $S=\emptyset$, then $|\mathcal{P}_S(n)|=2^{n-1}$ and $p_\emptyset(n)=1$.

Figures (4)

  • Figure 1: Roots of $p_{\{2,10\}}(z)$.
  • Figure 2: Roots of $p_{\{4,7,15\}}(z)$.
  • Figure 3: Roots of $p_{\{3,5,8,14\}}(z)$.
  • Figure 4: Roots of $p_{\{4,6,9,15\}}(z)$.

Theorems & Definitions (76)

  • Theorem \oldthetheorem: billey
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Conjecture \oldthetheorem: billey
  • Conjecture \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • ...and 66 more