The most frequent peak set of a random permutation

Abstract. Given a subset $S\subseteq\mathbb{P}$, let $P(S;n)$ be the number of permutations in the symmetric group of ${1,2,...,n}$ that have peak set $S$. We prove a recent conjecture due to Billey, Burdzy and Sagan, which determines the sets that maximize $P(S;n)$, where $S$ ranges over all subsets of ${1,2,...,n}$.
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