{"paper":{"title":"Elementary formulas for integer partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mohamed El Bachraoui","submitted_at":"2010-04-27T16:14:56Z","abstract_excerpt":"In this note we will give various exact formulas for functions on integer partitions including the functions $p(n)$ and $p(n,k)$ of the number of partitions of $n$ and the number of such partitions into exactly $k$ parts respectively. For instance, we shall prove that $$ p(n) = \\sum_{d|n} \\sum_{k=1}^{d} \\sum_{i_0 =1}^{\\lfloor d/k \\rfloor} \\sum_{i_1 =i_0}^{\\lfloor\\frac{d- i_0}{k-1} \\rfloor} \\sum_{i_2 =i_1}^{\\lfloor\\frac{d- i_0 - i_1}{k-2} \\rfloor} ... \\sum_{i_{k-3}=i_{k-4}}^{\\lfloor\\frac{n- i_0 - i_1-i_2- ...-i_{k-4}}{3} \\rfloor} \\sum_{c|(d,i_0,i_1,i_2,...,i_{k-3})} \\mu(c) (\\lfloor \\frac{d-i_0-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.4849","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"}