Conference abstracts

Session S06 - Algebraic Combinatorics

July 26, 16:00 ~ 16:25

## A proof of the peak polynomial positivity conjecture

### Pamela E. Harris

### Williams College, United States - pamela.e.harris@williams.edu

Given a permutation $\pi=\pi_1\pi_2\cdots \pi_n \in \mathfrak{S}_n$, 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 $P_S(n)=\{\pi\in\mathfrak{S}_n:P(\pi)=S\}$. 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 $p_S(x)$ depending on $S$. They gave a recursive formula for $p_S(n)$ involving an alternating sum, and they conjectured that the coefficients of $p_S(x)$ expanded in a binomial coefficient basis centered at $\max(S)$ are all nonnegative. In this talk we will share a different recursive formula for $p_S(n)$ without alternating sums, and we use this recursion to prove that their conjecture is true.

Joint work with Alexander Diaz-Lopez, Swarthmore College, Erik Insko, Florida Gulf Coast University and Mohamed Omar, Harvey Mudd College.