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

Abstract

The Erd\H os-Rényi 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-Rényi law for the longest consecutive monotone block in a random permutation.

A corrected version is under preparation