{"paper":{"title":"On the limiting distribution of some numbers of crossings in set partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anisse Kasraoui","submitted_at":"2013-01-28T13:25:22Z","abstract_excerpt":"We study the asymptotic distribution of the two following combinatorial parameters: the number of arc crossings in the linear representation, ${\\mathrm cr^{(\\ell)}$, and the number of chord crossings in the circular representation, ${\\mathrm cr^{(c)}$, of a random set partition. We prove that, for $k\\leq n/(2\\,\\log n)$ (resp., ${k=o(\\sqrt{n})}$), the distribution of the parameter ${\\mathrm cr^{(\\ell)}$ (resp., ${\\mathrm cr^{(c)}$) taken over partitions of $[n]:=\\{1,2,...,n\\}$ into $k$ blocks is, after standardization, asymptotically Gaussian as $n$ tends to infinity. We give exact and asymptot"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.6540","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"}