{"paper":{"title":"The number of s-separated k-sets in various circles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adri\\'an Pastine, Emiliano J.J. Estrugo","submitted_at":"2018-05-03T22:33:57Z","abstract_excerpt":"This article studies the number of ways of selecting $k$ objects arranged in $p$ circles of sizes $n_1,\\ldots,n_p$ such that no two selected ones have less than $s$ objects between them. If $n_i\\geq sk+1$ for all $1\\leq i \\leq p$, this number is shown to be $\\frac{n_1+\\ldots+n_p}{k}\\binom{n_1+\\ldots+n_p-sk-1}{k-1}$. A combinatorial proof of this claim is provided, and some nice combinatorial formulas are derived."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.01562","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"}