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

TL;DR

This paper extends the Erdős-Rényi law to the context of random permutations, establishing the asymptotic behavior of the longest consecutive monotone block within such permutations.

Contribution

It introduces a novel Erdős-Rényi law for the longest consecutive monotone block in random permutations, expanding the law's applicability beyond i.i.d. sequences.

Findings

Proves an Erdős-Rényi law for monotone blocks in permutations

Characterizes the asymptotic length of the longest monotone run

Extends classical results to permutation-based models

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.