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- W4300514944 abstract "We say that a permutation $pi=pi_1pi_2cdots pi_n in mathfrak{S}_n$ has a peak at index $i$ if $pi_{i-1} < pi_i > pi_{i+1}$. Let $mathcal{P}(pi)$ denote the set of indices where $pi$ has a peak. Given a set $S$ of positive integers, we define $mathcal{P}_S(n)={piinmathfrak{S}_n:mathcal{P}(pi)=S}$. In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers $S$ and sufficiently large $n$, $| mathcal{P}_S(n)|=p_S(n)2^{n-|S|-1}$ where $p_S(x)$ is a polynomial depending on $S$. They gave a recursive formula for $p_S(x)$ involving an alternating sum, and they conjectured that the coefficients of $p_S(x)$ expanded in a binomial coefficient basis centered at $max(S)$ are all nonnegative. In this paper we introduce a new recursive formula for $|mathcal{P}_S(n)|$ without alternating sums, and we use this recursion to prove that their conjecture is true." @default.
- W4300514944 created "2022-10-03" @default.
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- W4300514944 date "2016-05-05" @default.
- W4300514944 modified "2023-10-16" @default.
- W4300514944 title "A proof of the peak polynomial positivity conjecture" @default.
- W4300514944 doi "https://doi.org/10.48550/arxiv.1605.01708" @default.
- W4300514944 hasPublicationYear "2016" @default.
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