Factoring Peak Polynomials

Let Sn be the symmetric group of permutations π = π1π2 · · ·πn of {1, 2, . . . , n}. An index i of π is a peak if πi−1 < πi > πi+1, and we let P (π) denote the set of peaks of π. Given any set S of positive integers, we define PS(n) = {π ∈ Sn : P (π) = S}. Burdzy, Sagan, and the first author showed that for all fixed subsets of positive integers S and sufficiently large n we have |PS(n)| = pS(n)2 for some polynomial pS(x) depending on S. It is conjectured that the coefficients of pS(x) expanded in a binomial coefficient basis centered at max(S) are all positive, and we show that this is a consequence of a stronger conjecture that bounds the modulus of the zeros of pS(x). Our main results give an explicit formula for peak polynomials in the binomial basis centered at 0, show that all peaks are zeros of pS(x), and that 0, 1, 2, . . . , ir are zeros of pS(x) for any ir ∈ S if ir+1 − ir is odd. Additionally, we enumerate |PS(n)| using alternating permutations for all peak sets S.