# Permutations, peaks, polynomials, and a positivity conjecture

Posted on: October 17, 2017

# Colloquium

### Pamela Harris

Williams College

### Permutations, peaks, polynomials, and a positivity conjecture

### Friday, 10/27/17, 4pm, room 016.

From the basic ordering of n objects, solving a rubik’s cube, and establishing the unsolvability of the general quintic via radicals, permutations have played many important roles in mathematics. In this talk, we present some recent results related to the concept of peaks of permutations. A permutation π=π_{1}π_{2}⋯ π_{n} ∈ 𝒮_{n}, is said to have a peak at i if π_{i-1} < π_{i} > π_{i+1}. We let P(π)denote the set of peaks of π and, given any 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 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) called the peak polynomial of S. Billey, Burdzy, and Sagan conjectured that the coefficients of p_{S}(x) expanded in a binomial coefficient basis centered at max(S) are all nonnegative. We end this talk by sharing a new recursive formula for computing peak polynomials, which we then use to prove that their “positivity conjecture” is true.

Pamela is an assistant professor at Williams College who works in algebraic combinatorics. She is co-founder of the website http://lathisms.org/, featuring the accomplishments of Latinos and Hispanics in mathematics. See also this feature: http://vanguardstem.