{"paper":{"title":"The Twelvefold way, the non-intersecting circles problem, and partitions of multisets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniel Yaqubi, Madjid Mirzavaziri, Toufik Mansour","submitted_at":"2015-01-08T22:44:20Z","abstract_excerpt":"Let $n$ be a non-negative integer and $A=\\{a_1,\\ldots,a_k\\}$ be a multi-set with $k$ not necessarily distinct members, where $a_1\\leqslant\\ldots\\leqslant a_k$. We denote by $\\Delta(n,A)$ the number of ways to partition $n$ as the form $a_1x_1+\\ldots+a_kx_k$, where $x_i$'s are distinct positive integers and $x_i< x_{i+1}$ whenever $a_i=a_{i+1}$. We give a recursive formula for $\\Delta(n,A)$ and some explicit formulas for some special cases. Using this notion we solve the non-intersecting circles problem which asks to evaluate the number of ways to draw $n$ non-intersecting circles in a plane re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.01997","kind":"arxiv","version":2},"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"}