{"paper":{"title":"Distinct Partial Sums in Cyclic Groups: Polynomial Method and Constructive Approaches","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jacob Hicks, John. R. Schmitt, M. A. Ollis","submitted_at":"2018-09-07T21:35:28Z","abstract_excerpt":"Let $(G,+)$ be an abelian group and consider a subset $A \\subseteq G$ with $|A|=k$. Given an ordering $(a_1, \\ldots, a_k)$ of the elements of $A$, define its {\\em partial sums} by $s_0 = 0$ and $s_j = \\sum_{i=1}^j a_i$ for $1 \\leq j \\leq k$. We consider the following conjecture of Alspach: For any cyclic group $\\Z_n$ and any subset $A \\subseteq \\Z_n \\setminus \\{0\\}$ with $s_k \\neq 0$, it is possible to find an ordering of the elements of $A$ such that no two of its partial sums $s_i$ and $s_j$ are equal for $0 \\leq i < j \\leq k$. We show that Alspach's Conjecture holds for prime $n$ when $k \\g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.02684","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":""},"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"}