Permutation Patterns of the Iterated Syracuse Function
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866912044178997248 |
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| author | Nathanson, Melvyn B. |
| author_facet | Nathanson, Melvyn B. |
| contents | Let $Ω$ be the set of odd positive integers and let $S:Ω\rightarrow Ω$ be the Syracuse function. It is proved that, for every permutation $σ$ of $(1,2,3)$, the set of triples of the form $(m,S(m),S^2(m))$ with permutation pattern $σ$ has positive density, and these densities are computed. However, there exist permutations $τ$ of $(1,2,3,4)$ such that no quadruple $(m,S(m), S^2(m), S^3(m))$ has permutation pattern $τ$. This implies the nonexistence of certain permutation patterns of $n$-tuples $(m,S(m),\ldots, S^{n-1}(m))$ for all $n \geq 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_00644 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Permutation Patterns of the Iterated Syracuse Function Nathanson, Melvyn B. Number Theory Combinatorics 11A25, 11B83, 11B75, 11D04, 11N56, 05A05, 37P05 Let $Ω$ be the set of odd positive integers and let $S:Ω\rightarrow Ω$ be the Syracuse function. It is proved that, for every permutation $σ$ of $(1,2,3)$, the set of triples of the form $(m,S(m),S^2(m))$ with permutation pattern $σ$ has positive density, and these densities are computed. However, there exist permutations $τ$ of $(1,2,3,4)$ such that no quadruple $(m,S(m), S^2(m), S^3(m))$ has permutation pattern $τ$. This implies the nonexistence of certain permutation patterns of $n$-tuples $(m,S(m),\ldots, S^{n-1}(m))$ for all $n \geq 4$. |
| title | Permutation Patterns of the Iterated Syracuse Function |
| topic | Number Theory Combinatorics 11A25, 11B83, 11B75, 11D04, 11N56, 05A05, 37P05 |
| url | https://arxiv.org/abs/2308.00644 |