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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2605.29011 |
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| _version_ | 1866916056583372800 |
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| author | Gaiser, Collier Horn, Paul |
| author_facet | Gaiser, Collier Horn, Paul |
| contents | A sequence of positive integers $(a_1,a_2,\ldots,a_k)$ is called $\ell$-additive if $a_1+a_2+\cdots+a_k=\ell a_1$ or $\ell a_k$. In this paper, we prove that for all $k\geq3$, if $n$ is sufficiently large, then every permutation of $\{1,2,\ldots,n\}$ has a 2-additive subsequence of length $k$. We also provide polynomial bounds for the smallest $n$ such that every permutation of $\{1,2,\ldots,n\}$ has a 2-additive subsequence of length $k$. When only monotone subsequences are considered, we show that $18$ is the smallest $n$ such that every permutation of $\{1,2,\ldots,n\}$ has a monotone 2-additive subsequence of length three. Strong bounds are obtained for the minimum number of $\ell$-additive subsequences of any length, as well as monotone $2$-additive subsequences of length three. Using techniques in arithmetic Ramsey theory, we also show similar results for products and inverse sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29011 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Subsequence Sums in Permutations Gaiser, Collier Horn, Paul Combinatorics Number Theory 05A05, 05D10 A sequence of positive integers $(a_1,a_2,\ldots,a_k)$ is called $\ell$-additive if $a_1+a_2+\cdots+a_k=\ell a_1$ or $\ell a_k$. In this paper, we prove that for all $k\geq3$, if $n$ is sufficiently large, then every permutation of $\{1,2,\ldots,n\}$ has a 2-additive subsequence of length $k$. We also provide polynomial bounds for the smallest $n$ such that every permutation of $\{1,2,\ldots,n\}$ has a 2-additive subsequence of length $k$. When only monotone subsequences are considered, we show that $18$ is the smallest $n$ such that every permutation of $\{1,2,\ldots,n\}$ has a monotone 2-additive subsequence of length three. Strong bounds are obtained for the minimum number of $\ell$-additive subsequences of any length, as well as monotone $2$-additive subsequences of length three. Using techniques in arithmetic Ramsey theory, we also show similar results for products and inverse sums. |
| title | Subsequence Sums in Permutations |
| topic | Combinatorics Number Theory 05A05, 05D10 |
| url | https://arxiv.org/abs/2605.29011 |