{"paper":{"title":"On the Golomb-Dickman constant under Ewens sampling","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cond-mat.stat-mech","math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Jos\\'e Ricardo G. Mendon\\c{c}a, Luis Jehiel Negret","submitted_at":"2026-03-24T13:21:33Z","abstract_excerpt":"We define a generalized Golomb-Dickman constant $\\lambda_{\\theta}$ as the limiting expected proportion of the longest cycle in random permutations under the Ewens measure with parameter $\\theta > 0$. Exploiting the independence properties of Kingman's Poisson process construction of the Poisson-Dirichlet distribution, we obtain an explicit integral representation for $\\lambda_{\\theta}$ in terms of the exponential integral. The dependence of $\\lambda_{\\theta}$ on $\\theta$ reflects the transition between regimes dominated by long cycles (small $\\theta$) and those with many small cycles (large $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.23175","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.23175/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}