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| Formato: | Preprint |
| Publicado: |
2022
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| Acceso en línea: | https://arxiv.org/abs/2211.04451 |
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| _version_ | 1866917731150856192 |
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| author | Adenwalla, Sarosh |
| author_facet | Adenwalla, Sarosh |
| contents | A permutation of the integers avoiding monotone arithmetic progressions of length $6$ was constructed in (Geneson, 2018). We improve on this by constructing a permutation of the integers avoiding monotone arithmetic progressions of length $5$. We also construct permutations of the integers and the positive integers that improve on previous upper and lower density results. In (Davis et al. 1977) they constructed a doubly infinite permutation of the positive integers that avoids monotone arithmetic progressions of length $4$. We construct a doubly infinite permutation of the integers avoiding monotone arithmetic progressions of length $5$. A permutation of the positive integers that avoided monotone arithmetic progressions of length $4$ with odd common difference was constructed in (LeSaulnier and Vijay, 2011). We generalise this result and show that for each $k\geq 1$, there exists a permutation of the positive integers that avoids monotone arithmetic progressions of length $4$ with common difference not divisible by $2^k$. In addition, we specify the structure of permutations of $[1,n]$ that avoid length $3$ monotone arithmetic progressions mod $n$ as defined in (Davis et al. 1977) and provide an explicit construction for a multiplicative result on permutations that avoid length $k$ monotone arithmetic progressions mod $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_04451 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Avoiding Monotone Arithmetic Progressions in Permutations of Integers Adenwalla, Sarosh Combinatorics 11B25 A permutation of the integers avoiding monotone arithmetic progressions of length $6$ was constructed in (Geneson, 2018). We improve on this by constructing a permutation of the integers avoiding monotone arithmetic progressions of length $5$. We also construct permutations of the integers and the positive integers that improve on previous upper and lower density results. In (Davis et al. 1977) they constructed a doubly infinite permutation of the positive integers that avoids monotone arithmetic progressions of length $4$. We construct a doubly infinite permutation of the integers avoiding monotone arithmetic progressions of length $5$. A permutation of the positive integers that avoided monotone arithmetic progressions of length $4$ with odd common difference was constructed in (LeSaulnier and Vijay, 2011). We generalise this result and show that for each $k\geq 1$, there exists a permutation of the positive integers that avoids monotone arithmetic progressions of length $4$ with common difference not divisible by $2^k$. In addition, we specify the structure of permutations of $[1,n]$ that avoid length $3$ monotone arithmetic progressions mod $n$ as defined in (Davis et al. 1977) and provide an explicit construction for a multiplicative result on permutations that avoid length $k$ monotone arithmetic progressions mod $n$. |
| title | Avoiding Monotone Arithmetic Progressions in Permutations of Integers |
| topic | Combinatorics 11B25 |
| url | https://arxiv.org/abs/2211.04451 |