Coefficients and 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.

20 pages, 5 figures and tables, final version with minor changes suggested by the referees

Cited by in corpus (2)

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