Permutations, Peaks Polynomials, and a Positivity Conjecture
Professor Pamela E. Harris
Williams College
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∈ Gn 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 ∈ Gn : 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|-1for 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 E. Harris is an Assistant Professor in the department of Mathematics and Statistics at Williams College. Her research interests are in algebra and combinatorics, particularly as these subjects relate to the representation theory of Lie algebras. She received her B.S. from Marquette University, and M.S. and Ph.D. in mathematics from the University of Wisconsin-Milwaukee. She often co-organizes research symposia sessions and professional development sessions for the national SACNAS conference. She was an MAA Project NExT (New Experiences in Teaching) Fellow from 2012-2013, and is a member of the AMS, AWM, MAA, NAM and SACNAS.
