A proof of the peak polynomial positivity conjecture
Mohamed Bashir Omar2017, Journal of Combinatorial Theory, Series A
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Abstract
We say that a permutation π = π 1 π 2 • • • π n ∈ S n has a peak at index i if π i−1 < π i > π i+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 ∈ S n : P(π) = S}. In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers S and sufficiently large n, |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 |P S (n)| without alternating sums, and we use this recursion to prove that their conjecture is true.
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