Cheyne Homberger

The minimum Manhattan distance and minimum jump of permutations

with Simon Blackburn Peter Winkler

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, $\br(\pi)$, between any pair of dots. The paper computes the expected value of $\br(\pi)$ 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 expected value of $\mj(\pi)$, and the asymptotic probability that $\mj(\pi)\geq d+1$ for any constant $d$.