## Sara Billey ; Matthew Fahrbach ; Alan Talmage - Factoring peak polynomials

dmtcs:2478 - Discrete Mathematics & Theoretical Computer Science, January 1, 2015, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) - https://doi.org/10.46298/dmtcs.2478
Factoring peak polynomials

Authors: Sara Billey 1; Matthew Fahrbach 2; Alan Talmage 2

• 1 Department of Mathematics [Seattle]
• 2 Department of Mathematics

Given a permutation $\pi=\pi_1\pi_2\cdots \pi_n \in S_n$, we say an index $i$ is a peak if $\pi_{i-1} < \pi_i > \pi_{i+1}$. Let $P(\pi)$ denote the set of peaks of $\pi$. Given any set $S$ of positive integers, define ${P_S(n)=\{\pi\in S_n:P(\pi)=S\}}$. Billey-Burdzy-Sagan showed that for all fixed subsets of positive integers $S$ and sufficiently large $n$, $|P_S(n)|=p_S(n)2^{n-|S|-1}$ 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.

Volume: DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
Section: Proceedings
Published on: January 1, 2015
Imported on: November 21, 2016
Keywords: binomial coefficient,combinatorics,peak,permutation,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
Source : OpenAIRE Graph
• REU 2011: Inverse Problems for Electrical Networks; Funder: National Science Foundation; Code: 1062253
• Combinatorial and Algebraic Aspects of Varieties; Funder: National Science Foundation; Code: 1101017