Coefficients and Roots of Peak Polynomials

ABSTRACT

Given a permutation , we say an index i is a peak if π_{i − 1} < π_i > π_{i + 1}. Let P(π) denote the set of peaks of π. Given any set S of positive integers, define . Billey–Burdzy–Sagan showed that for all fixed subsets of positive integers S and sufficiently large n, for some polynomial p_S(x) depending on S. They conjectured that the coefficients of p_S(x) expanded in a binomial coefficient basis centered at max (S) are all positive. We show that this is a consequence of a stronger conjecture that bounds the modulus of the roots of p_S(x). Furthermore, we give an efficient explicit formula for peak polynomials in the binomial basis centered at 0, which we use to identify many integer roots of peak polynomials along with certain inequalities and identities.

Keywords:

• binomial coefficient
• combinatorics
• peak
• permutation

2000 AMS Subject Classification:

• 05A05

Acknowledgments

We would like to thank Jim Morrow first and foremost for organizing the University of Washington Mathematics REU for over 25 years. We also would like to thank Ben Braun, Tom Edwards, Richard Ehrenborg, Noam Elkies, Daniel Hirsbrunner, Jerzy Jaromczyk, Beth Kelly, and Austin Tran for their discussions with us about various results in this paper. We credit SageMath [Stein 13] and the Online Encyclopedia of Integer Sequences [OEIS 13] for assisting with our research. Finally, we would like to thank the anonymous reviewers for their helpful suggestions.

Funding

Support for this work was provided by the National Science Foundation under grants DMS-1062253 and DMS-1101017, and the University of Washington Mathematics REU 2013 and 2014.