**The minimum Manhattan distance and minimum jump of permutations.** / Blackburn, Simon; Homberger, Cheyne; Winkler, Peter.

Research output: Contribution to journal › Article

Published

**The minimum Manhattan distance and minimum jump of permutations.** / Blackburn, Simon; Homberger, Cheyne; Winkler, Peter.

Research output: Contribution to journal › Article

Blackburn, S, Homberger, C & Winkler, P 2019, 'The minimum Manhattan distance and minimum jump of permutations' *Journal of Combinatorial Theory, Series A*, vol. 161, pp. 364-286. https://doi.org/10.1016/j.jcta.2018.09.002

Blackburn, S., Homberger, C., & Winkler, P. (2019). The minimum Manhattan distance and minimum jump of permutations. *Journal of Combinatorial Theory, Series A*, *161*, 364-286. https://doi.org/10.1016/j.jcta.2018.09.002

Blackburn S, Homberger C, Winkler P. The minimum Manhattan distance and minimum jump of permutations. Journal of Combinatorial Theory, Series A. 2019 Jan;161:364-286. https://doi.org/10.1016/j.jcta.2018.09.002

@article{10d1f611d78f4be19364ccb1a7e40320,

title = "The minimum Manhattan distance and minimum jump of 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, $\br(\pi)$, between any pair of dots. The paper computes the expected value (and higher moments) 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 asymptotic moments of $\mj(\pi)$, and the asymptotic probability that $\mj(\pi)\geq d+1$ for any constant $d$.",

author = "Simon Blackburn and Cheyne Homberger and Peter Winkler",

year = "2019",

month = "1",

doi = "10.1016/j.jcta.2018.09.002",

language = "English",

volume = "161",

pages = "364--286",

journal = "Journal of Combinatorial Theory, Series A",

issn = "0097-3165",

publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - The minimum Manhattan distance and minimum jump of permutations

AU - Blackburn, Simon

AU - Homberger, Cheyne

AU - Winkler, Peter

PY - 2019/1

Y1 - 2019/1

N2 - 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 (and higher moments) 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 asymptotic moments of $\mj(\pi)$, and the asymptotic probability that $\mj(\pi)\geq d+1$ for any constant $d$.

AB - 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 (and higher moments) 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 asymptotic moments of $\mj(\pi)$, and the asymptotic probability that $\mj(\pi)\geq d+1$ for any constant $d$.

U2 - 10.1016/j.jcta.2018.09.002

DO - 10.1016/j.jcta.2018.09.002

M3 - Article

VL - 161

SP - 364

EP - 286

JO - Journal of Combinatorial Theory, Series A

JF - Journal of Combinatorial Theory, Series A

SN - 0097-3165

ER -