# The minimum Manhattan distance and minimum jump of permutations

## TL;DR

This paper analyzes the expected minimum Manhattan distance and jump in random permutations, providing asymptotic results and settling a conjecture related to permutation pattern distances.

## Contribution

It computes the expected values and higher moments of the minimum Manhattan distance and jump in permutations, and proves a conjecture on their asymptotic probabilities.

## Key findings

- Expected value of minimum Manhattan distance as n→∞
- Asymptotic probability that distance exceeds a fixed d
- Asymptotic moments of minimum jump in permutations

## Abstract

Let $\pi$ be a permutation of $\{1,2,\ldots,n\}$. If we identify a permutation with its graph, namely the set of $n$ dots at positions $(i,\pi(i))$, it is natural to consider the minimum $L^1$ (Manhattan) distance, $d(\pi)$, between any pair of dots. The paper computes the expected value (and higher moments) of $d(\pi)$ when $n\rightarrow\infty$ and $\pi$ is chosen uniformly, and settles a conjecture of Bevan, Homberger and Tenner (motivated by permutation patterns), showing that when $d$ is fixed and $n\rightarrow\infty$, the probability that $d(\pi)\geq d+2$ tends to $e^{-d^2 - d}$. The minimum jump $mj(\pi)$ of $\pi$, defined by $mj(\pi)=\min_{1\leq i\leq n-1} |\pi(i+1)-\pi(i)|$, is another natural measure in this context. The paper computes the asymptotic moments of $mj(\pi)$, and the asymptotic probability that $mj(\pi)\geq d+1$ for any constant $d$.