{"paper":{"title":"Some results on ordered and unordered factorization of a positive integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniel Yaqubi, Madjid Mirzavaziri","submitted_at":"2014-12-01T09:33:02Z","abstract_excerpt":"As a well-known enumerative problem, the number of solutions of the equation $m=m_1+...+m_k$ with $m_1\\leqslant...\\leqslant m_k$ in positive integers is $\\Pi(m,k)=\\sum_{i=0}^k\\Pi(m-k,i)$ and $\\Pi$ is called the additive partition function. In this paper, we give a recursive formula for the so-called multiplicative partition function $\\mu_1(m,k):=$ the number of solutions of the equation $m=m_1... m_k$ with $m_1\\leqslant...\\leqslant m_k$ in positive integers. In particular, using an elementary proof, we give an explicit formula for the cases $k=1,2,3,4$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.0392","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"}