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

Anant Godbole

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arXiv · 2602.18970

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

Anant Godbole

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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.

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Anant Godbole. 2026-02-21. An Erd\H os R\'enyi Law for the Longest Consecutive Monotone Block in a Random Permutation. https://arxiv.org/abs/2602.18970

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