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A proof of the peak polynomial positivity conjecture

We say that a permutation $π=π_1π_2\cdots π_n \in \mathfrak{S}_n$ has a peak at index $i$ if $π_{i-1} < π_i > π_{i+1}$. Let $\mathcal{P}(π)$ denote the set of indices where $π$ has a peak. Given a set $S$ of positive integers, we define $\mathcal{P}_S(n)=\{π\in\mathfrak{S}_n:\mathcal{P}(π)=S\}$. 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.