Coefficients and roots of peak polynomials

TL;DR

This paper studies peak polynomials associated with permutation peaks, proving a conjecture about their roots and coefficients, and providing explicit formulas and roots for these polynomials.

Contribution

It proves that the positivity of polynomial coefficients follows from bounds on their roots and offers explicit formulas and root characterizations for peak polynomials.

Findings

Coefficients in a binomial basis are positive if roots are bounded.

Explicit formulas for peak polynomials in the binomial basis centered at 0.

Identification of many integer roots of peak polynomials.

Abstract

Given a permutation $\pi=\pi_1\pi_2\cdots \pi_n \in \mathfrak{S}_n$, we say an index $i$ is a peak if $\pi_{i-1} < \pi_i > \pi_{i+1}$. Let $P(\pi)$ denote the set of peaks of $\pi$. Given any set $S$ of positive integers, define ${\mathcal{P}_S(n)=\{\pi\in \mathfrak{S}_n:P(\pi)=S\}}$. 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.