{"paper":{"title":"A formula for the partition function that \"counts\"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrew V. Sills, Yuriy Choliy","submitted_at":"2018-11-22T23:27:36Z","abstract_excerpt":"We derive a combinatorial multisum expression for the number $D(n,k)$ of partitions of $n$ with Durfee square of order $k$. An immediate corollary is therefore a combinatorial formula for $p(n)$, the number of partitions of $n$. We then study $D(n,k)$ as a quasipolynomial. We consider the natural polynomial approximation $\\tilde{D}(n,k)$ to the quasipolynomial representation of $D(n,k)$. Numerically, the sum $\\sum_{1\\leq k \\leq \\sqrt{n}} \\tilde{D}(n,k)$ appears to be extremely close to the initial term of the Hardy--Ramanujan--Rademacher convergent series for $p(n)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.09327","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"}