{"paper":{"title":"Distributions of Hook Lengths Divisible by Two or Three","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.PR"],"primary_cat":"math.NT","authors_text":"Hamilton Wan, Hannah Lang, Nancy Xu","submitted_at":"2022-07-13T19:19:29Z","abstract_excerpt":"For fixed $t = 2$ or $3$, we investigate the statistical properties of $\\{Y_t(n)\\}$, the sequence of random variables corresponding to the number of hook lengths divisible by $t$ among the partitions of $n$. We characterize the support of $Y_t(n)$ and show, in accordance with empirical observations, that the support is vanishingly small for large $n$. Moreover, we demonstrate that the nonzero values of the mass functions of $Y_2(n)$ and $Y_3(n)$ approximate continuous functions. Finally, we prove that although the mass functions fail to converge, the cumulative distribution functions of $\\{Y_2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.06486","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/2207.06486/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"}