{"paper":{"title":"Subsequence Sums in Permutations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Collier Gaiser, Paul Horn","submitted_at":"2026-05-27T19:08:25Z","abstract_excerpt":"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-add"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.29011","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.29011/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}