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Factoring peak polynomials

Abstract : Given a permutation $\pi=\pi_1\pi_2\cdots \pi_n \in S_n$, 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 ${P_S(n)=\{\pi\in S_n:P(\pi)=S\}}$. Billey-Burdzy-Sagan showed that for all fixed subsets of positive integers $S$ and sufficiently large $n$, $|P_S(n)|=p_S(n)2^{n-|S|-1}$ 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.
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https://hal.archives-ouvertes.fr/hal-01337803
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Submitted on : Monday, June 27, 2016 - 3:22:51 PM
Last modification on : Friday, June 28, 2019 - 2:24:04 PM

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  • HAL Id : hal-01337803, version 1

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Sara Billey, Matthew Fahrbach, Alan Talmage. Factoring peak polynomials. 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), Jul 2015, Daejeon, South Korea. pp.441-452. ⟨hal-01337803⟩

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