An Erd\H os R\'enyi Law for the Longest Consecutive Monotone Block in a Random Permutation

Abstract

The Erd\H os-R\'enyi law states that given a sequence $\{X_j\}_{j=1}^\infty$ of i.i.d.~($p$) coin-tosses, the longest run $L_n$ of heads in the first $n$ coin tosses approaches $\log_{1/p}n$ almost surely. In this paper we explore a formulation of this result in the case of random permutations and prove an Erd\H os-R\'enyi law for the longest consecutive monotone block in a random permutation.