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=π1π2⋯ πn ∈ 𝒮n​​, is said to have a peak at i​​ if πi-1 < πi > πi+1​​. We let P(π)​​denote the set of peaks of π​​ and, given any set S​​ of positive integers, we define PS(n)={π∈𝒮n:P(π)=S}. In 2013 Billey, Burdzy, and Sagan showed that for all fixed subsets of positive integers S​​ and sufficiently large n, |PS(n)|=pS(n)2n-|S|-1​​ for some polynomial pS(x)​​ called the peak polynomial of S​​. Billey, Burdzy, and Sagan conjectured that the coefficients of pS(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.com/wcwinstem-pamela-e-harris-ph-d/.