{"paper":{"title":"Permutations with small maximal $k$-consecutive sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Akihiro Higashitani, Kazuki Kurimoto","submitted_at":"2018-01-01T09:20:19Z","abstract_excerpt":"Let $n$ and $k$ be positive integers with $n>k$. Given a permutation $(\\pi_1,\\ldots,\\pi_n)$ of integers $1,\\ldots,n$, we consider $k$-consecutive sums of $\\pi$, i.e., $s_i:=\\sum_{j=0}^{k-1}\\pi_{i+j}$ for $i=1,\\ldots,n$, where we let $\\pi_{n+j}=\\pi_j$. What we want to do in this paper is to know the exact value of $$\\mathrm{msum}(n,k):=\\min\\left\\{\\max\\{s_i : i=1,\\ldots,n\\} -\\frac{k(n+1)}{2}: \\pi \\in S_n\\right\\},$$ where $S_n$ denotes the set of all permutations of $1,\\ldots,n$. In this paper, we determine the exact values of $\\mathrm{msum}(n,k)$ for some particular cases of $n$ and $k$. As a co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.00416","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}