A proof of the peak polynomial positivity conjecture

We say that a permutation π=π1π2⋯πn∈Sn has a peak at index i if πi−1<πi>πi+1. Let P(π) denote the set of indices where π has a peak. Given a set S of positive integers, we define P(S;n)={π∈Sn:P(π)=S}. In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers S and sufficiently large n, |P(S;n)|=pS(n)2n−|S|−1 where pS(x) is a polynomial depending on S. They proved this by establishing a recursive formula for pS(x) involving an alternating sum, and they conjectured that the coefficients of pS(x) expanded in a binomial coefficient basis centered at max⁡(S) are all nonnegative. In this paper we introduce a new recursive formula for |P(S;n)| without alternating sums and we use this recursion to prove that their conjecture is true.