Coefficients and roots of peak polynomials
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2014
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| _version_ | 1866911902467096576 |
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| author | Billey, Sara Fahrbach, Matthew Talmage, Alan |
| author_facet | Billey, Sara Fahrbach, Matthew Talmage, Alan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1410_8506 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Coefficients and roots of peak polynomials Billey, Sara Fahrbach, Matthew Talmage, Alan Combinatorics 05C30 G.2.1 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. |
| title | Coefficients and roots of peak polynomials |
| topic | Combinatorics 05C30 G.2.1 |
| url | https://arxiv.org/abs/1410.8506 |