We say that a permutation π=π1π2⋯πn∈Snπ=π1π2⋯πn∈Sn has a peak at index i   if πi−1 πi+1πi−1 πi+1. Let P(π)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}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|P(S;n)|=pS(n)2n−|S|−1 where pS(x)pS(x) is a polynomial depending on S  . They proved this by establishing a recursive formula for pS(x)pS(x) involving an alternating sum, and they conjectured that the coefficients of pS(x)pS(x) expanded in a binomial coefficient basis centered at max⁡(S)max⁡(S) are all nonnegative. In this paper we introduce a new recursive formula for |P(S;n)||P(S;n)| without alternating sums and we use this recursion to prove that their conjecture is true.
['Alexander Diaz-Lopez', 'Pamela E. Harris', 'Erik K. Insko', 'Mohamed Omar']
A proof of the peak polynomial positivity conjecture