Permutations that Separate Close Elements, and Rectangle Packings in the Torus
Let $n$, $s$ and $k$ be positive integers. For distinct $i,j\in\mathbb{Z}_n$, define $||i,j||_n$ to be the distance between $i$ and $j$ when the elements of $\mathbb{Z}_n$ are written in a circle. So\[||i,j||_n=\min\{(i-j)\bmod n,(j-i)\bmod n\}.\]A permutation $\pi:\mathbb{Z}_n\rightarrow\mathbb {Z}...
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| Veröffentlicht in: | The Electronic journal of combinatorics 2024-11, Vol.31 (4) |
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| container_title | The Electronic journal of combinatorics |
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| creator | Blackburn, Simon R. Etzion, Tuvi |
| description | Let $n$, $s$ and $k$ be positive integers. For distinct $i,j\in\mathbb{Z}_n$, define $||i,j||_n$ to be the distance between $i$ and $j$ when the elements of $\mathbb{Z}_n$ are written in a circle. So\[||i,j||_n=\min\{(i-j)\bmod n,(j-i)\bmod n\}.\]A permutation $\pi:\mathbb{Z}_n\rightarrow\mathbb {Z}_n$ is $(s,k)$-clash-free if $||\pi(i),\pi(j)||_n\geq k$ whenever $||i,j||_n |
| doi_str_mv | 10.37236/12711 |
| format | Article |
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For distinct $i,j\in\mathbb{Z}_n$, define $||i,j||_n$ to be the distance between $i$ and $j$ when the elements of $\mathbb{Z}_n$ are written in a circle. So\[||i,j||_n=\min\{(i-j)\bmod n,(j-i)\bmod n\}.\]A permutation $\pi:\mathbb{Z}_n\rightarrow\mathbb {Z}_n$ is $(s,k)$-clash-free if $||\pi(i),\pi(j)||_n\geq k$ whenever $||i,j||_n<s$. So an $(s,k)$-clash-free permutation moves every pair of close elements (at distance less than $s$) to a pair of elements at large distance (at distance at least $k$). The notion of an $(s,k)$-clash-free permutation can be reformulated in terms of certain packings of $s\times k$ rectangles on an $n\times n$ torus. For integers $n$ and $k$ with $1\leq k<n$, let $\sigma(n,k)$ be the largest value of $s$ such that an $(s,k)$-clash-free permutation of $\mathbb{Z}_n$ exists. Strengthening a recent paper of Blackburn, which proved a conjecture of Mammoliti and Simpson, we determine the value of $\sigma(n,k)$ in all cases.</description><identifier>ISSN: 1077-8926</identifier><identifier>EISSN: 1077-8926</identifier><identifier>DOI: 10.37236/12711</identifier><language>eng</language><ispartof>The Electronic journal of combinatorics, 2024-11, Vol.31 (4)</ispartof><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,864,27924,27925</link.rule.ids></links><search><creatorcontrib>Blackburn, Simon R.</creatorcontrib><creatorcontrib>Etzion, Tuvi</creatorcontrib><title>Permutations that Separate Close Elements, and Rectangle Packings in the Torus</title><title>The Electronic journal of combinatorics</title><description>Let $n$, $s$ and $k$ be positive integers. For distinct $i,j\in\mathbb{Z}_n$, define $||i,j||_n$ to be the distance between $i$ and $j$ when the elements of $\mathbb{Z}_n$ are written in a circle. So\[||i,j||_n=\min\{(i-j)\bmod n,(j-i)\bmod n\}.\]A permutation $\pi:\mathbb{Z}_n\rightarrow\mathbb {Z}_n$ is $(s,k)$-clash-free if $||\pi(i),\pi(j)||_n\geq k$ whenever $||i,j||_n<s$. So an $(s,k)$-clash-free permutation moves every pair of close elements (at distance less than $s$) to a pair of elements at large distance (at distance at least $k$). The notion of an $(s,k)$-clash-free permutation can be reformulated in terms of certain packings of $s\times k$ rectangles on an $n\times n$ torus. For integers $n$ and $k$ with $1\leq k<n$, let $\sigma(n,k)$ be the largest value of $s$ such that an $(s,k)$-clash-free permutation of $\mathbb{Z}_n$ exists. Strengthening a recent paper of Blackburn, which proved a conjecture of Mammoliti and Simpson, we determine the value of $\sigma(n,k)$ in all cases.</description><issn>1077-8926</issn><issn>1077-8926</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNpNkLFOwzAURS0EEqXAN3hiIuBnx7E9oqgUpAoqKHP0YjslkDiV7Q78PagwMJ27nDscQi6B3QjFRXULXAEckRkwpQpteHX8b5-Ss5Q-GANujJyRp7WP4z5j7qeQaH7HTF_9DiNmT-thSp4uBj_6kNM1xeDoi7cZw3bwdI32sw_bRPvw43m6meI-nZOTDofkL_44J2_3i039UKyel4_13aqwUOpcOMt5h4AMoLJlpcoOhNfYaiaNk9qCM8CEksxWzrStldaVFjRzynDmQIo5ufr9tXFKKfqu2cV-xPjVAGsOFZpDBfENO39OHw</recordid><startdate>20241101</startdate><enddate>20241101</enddate><creator>Blackburn, Simon R.</creator><creator>Etzion, Tuvi</creator><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20241101</creationdate><title>Permutations that Separate Close Elements, and Rectangle Packings in the Torus</title><author>Blackburn, Simon R. ; Etzion, Tuvi</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c148t-dc22fa1a0116c4674f13e8ab8059d58c1d9103750c6d9bbc5cd4c180d7920d153</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Blackburn, Simon R.</creatorcontrib><creatorcontrib>Etzion, Tuvi</creatorcontrib><collection>CrossRef</collection><jtitle>The Electronic journal of combinatorics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Blackburn, Simon R.</au><au>Etzion, Tuvi</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Permutations that Separate Close Elements, and Rectangle Packings in the Torus</atitle><jtitle>The Electronic journal of combinatorics</jtitle><date>2024-11-01</date><risdate>2024</risdate><volume>31</volume><issue>4</issue><issn>1077-8926</issn><eissn>1077-8926</eissn><abstract>Let $n$, $s$ and $k$ be positive integers. For distinct $i,j\in\mathbb{Z}_n$, define $||i,j||_n$ to be the distance between $i$ and $j$ when the elements of $\mathbb{Z}_n$ are written in a circle. So\[||i,j||_n=\min\{(i-j)\bmod n,(j-i)\bmod n\}.\]A permutation $\pi:\mathbb{Z}_n\rightarrow\mathbb {Z}_n$ is $(s,k)$-clash-free if $||\pi(i),\pi(j)||_n\geq k$ whenever $||i,j||_n<s$. So an $(s,k)$-clash-free permutation moves every pair of close elements (at distance less than $s$) to a pair of elements at large distance (at distance at least $k$). The notion of an $(s,k)$-clash-free permutation can be reformulated in terms of certain packings of $s\times k$ rectangles on an $n\times n$ torus. For integers $n$ and $k$ with $1\leq k<n$, let $\sigma(n,k)$ be the largest value of $s$ such that an $(s,k)$-clash-free permutation of $\mathbb{Z}_n$ exists. Strengthening a recent paper of Blackburn, which proved a conjecture of Mammoliti and Simpson, we determine the value of $\sigma(n,k)$ in all cases.</abstract><doi>10.37236/12711</doi><oa>free_for_read</oa></addata></record> |
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| ispartof | The Electronic journal of combinatorics, 2024-11, Vol.31 (4) |
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| language | eng |
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| source | DOAJ Directory of Open Access Journals; EZB-FREE-00999 freely available EZB journals |
| title | Permutations that Separate Close Elements, and Rectangle Packings in the Torus |
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