{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:ZWXFEW3C3YWM2QOBHCCYF64PVM","short_pith_number":"pith:ZWXFEW3C","schema_version":"1.0","canonical_sha256":"cdae525b62de2ccd41c1388582fb8fab0189bc6af302f93ff7d2bca8f0875135","source":{"kind":"arxiv","id":"1911.03135","version":3},"attestation_state":"computed","paper":{"title":"The size of $t$-cores and hook lengths of random cells in random partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.PR","authors_text":"Arvind Ayyer, Shubham Sinha","submitted_at":"2019-11-08T09:07:10Z","abstract_excerpt":"Fix $t \\geq 2$. We first give an asymptotic formula for certain sums of the number of $t$-cores. We then use this result to compute the distribution of the size of the $t$-core of a uniformly random partition of an integer $n$. We show that this converges weakly to a gamma distribution after dividing by $\\sqrt{n}$. As a consequence, we find that the size of the $t$-core is of the order of $\\sqrt{n}$ in expectation. We then apply this result to show that the probability that $t$ divides the hook length of a uniformly random cell in a uniformly random partition equals $1/t$ in the limit. Finally"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1911.03135","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-11-08T09:07:10Z","cross_cats_sorted":["math.CO","math.NT"],"title_canon_sha256":"4b736c2c52a0fb56c7406879c4a4b5ac41197ae23006f5c37a424321d84d2433","abstract_canon_sha256":"8cd5c86a30e3323c839872861d1bfecea0414b53c587222b72c4ff83d458a5bc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:45:22.230518Z","signature_b64":"KsUiIDdC4f7RBf46Qo1Jxc3JjCDs/TOJrXO0+dc6W8bIINxuERDfaCV3Ul5kLoWlf518Bf1Y5x8Or4swRcyDAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cdae525b62de2ccd41c1388582fb8fab0189bc6af302f93ff7d2bca8f0875135","last_reissued_at":"2026-07-05T05:45:22.230142Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:45:22.230142Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The size of $t$-cores and hook lengths of random cells in random partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.PR","authors_text":"Arvind Ayyer, Shubham Sinha","submitted_at":"2019-11-08T09:07:10Z","abstract_excerpt":"Fix $t \\geq 2$. We first give an asymptotic formula for certain sums of the number of $t$-cores. We then use this result to compute the distribution of the size of the $t$-core of a uniformly random partition of an integer $n$. We show that this converges weakly to a gamma distribution after dividing by $\\sqrt{n}$. As a consequence, we find that the size of the $t$-core is of the order of $\\sqrt{n}$ in expectation. We then apply this result to show that the probability that $t$ divides the hook length of a uniformly random cell in a uniformly random partition equals $1/t$ in the limit. Finally"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.03135","kind":"arxiv","version":3},"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/1911.03135/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1911.03135","created_at":"2026-07-05T05:45:22.230198+00:00"},{"alias_kind":"arxiv_version","alias_value":"1911.03135v3","created_at":"2026-07-05T05:45:22.230198+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1911.03135","created_at":"2026-07-05T05:45:22.230198+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZWXFEW3C3YWM","created_at":"2026-07-05T05:45:22.230198+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZWXFEW3C3YWM2QOB","created_at":"2026-07-05T05:45:22.230198+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZWXFEW3C","created_at":"2026-07-05T05:45:22.230198+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZWXFEW3C3YWM2QOBHCCYF64PVM","json":"https://pith.science/pith/ZWXFEW3C3YWM2QOBHCCYF64PVM.json","graph_json":"https://pith.science/api/pith-number/ZWXFEW3C3YWM2QOBHCCYF64PVM/graph.json","events_json":"https://pith.science/api/pith-number/ZWXFEW3C3YWM2QOBHCCYF64PVM/events.json","paper":"https://pith.science/paper/ZWXFEW3C"},"agent_actions":{"view_html":"https://pith.science/pith/ZWXFEW3C3YWM2QOBHCCYF64PVM","download_json":"https://pith.science/pith/ZWXFEW3C3YWM2QOBHCCYF64PVM.json","view_paper":"https://pith.science/paper/ZWXFEW3C","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1911.03135&json=true","fetch_graph":"https://pith.science/api/pith-number/ZWXFEW3C3YWM2QOBHCCYF64PVM/graph.json","fetch_events":"https://pith.science/api/pith-number/ZWXFEW3C3YWM2QOBHCCYF64PVM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZWXFEW3C3YWM2QOBHCCYF64PVM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZWXFEW3C3YWM2QOBHCCYF64PVM/action/storage_attestation","attest_author":"https://pith.science/pith/ZWXFEW3C3YWM2QOBHCCYF64PVM/action/author_attestation","sign_citation":"https://pith.science/pith/ZWXFEW3C3YWM2QOBHCCYF64PVM/action/citation_signature","submit_replication":"https://pith.science/pith/ZWXFEW3C3YWM2QOBHCCYF64PVM/action/replication_record"}},"created_at":"2026-07-05T05:45:22.230198+00:00","updated_at":"2026-07-05T05:45:22.230198+00:00"}