{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:4PWYEDCBZXRVXSPXSIMHUXJQWI","short_pith_number":"pith:4PWYEDCB","canonical_record":{"source":{"id":"1508.07637","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-30T21:04:47Z","cross_cats_sorted":[],"title_canon_sha256":"c09d90ede7c8172d43670024a1f447feabcc22a493811baa8730a756bc1308c6","abstract_canon_sha256":"1af2faee046793c9facf254d6c47989000339ea84123e61404a08cc6099dd44d"},"schema_version":"1.0"},"canonical_sha256":"e3ed820c41cde35bc9f792187a5d30b213c458b5d352763517fec613b6a66b42","source":{"kind":"arxiv","id":"1508.07637","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.07637","created_at":"2026-05-18T01:34:07Z"},{"alias_kind":"arxiv_version","alias_value":"1508.07637v2","created_at":"2026-05-18T01:34:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.07637","created_at":"2026-05-18T01:34:07Z"},{"alias_kind":"pith_short_12","alias_value":"4PWYEDCBZXRV","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"4PWYEDCBZXRVXSPX","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"4PWYEDCB","created_at":"2026-05-18T12:29:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:4PWYEDCBZXRVXSPXSIMHUXJQWI","target":"record","payload":{"canonical_record":{"source":{"id":"1508.07637","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-30T21:04:47Z","cross_cats_sorted":[],"title_canon_sha256":"c09d90ede7c8172d43670024a1f447feabcc22a493811baa8730a756bc1308c6","abstract_canon_sha256":"1af2faee046793c9facf254d6c47989000339ea84123e61404a08cc6099dd44d"},"schema_version":"1.0"},"canonical_sha256":"e3ed820c41cde35bc9f792187a5d30b213c458b5d352763517fec613b6a66b42","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:07.088489Z","signature_b64":"w2Cv/gXaHm79rM5mvNARK+ZgW1J0Fj0MI9YlZglD0HNLxnHZ7q94BhQTxWNVQnNm8vND5yjmnznnk1atL4RvBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e3ed820c41cde35bc9f792187a5d30b213c458b5d352763517fec613b6a66b42","last_reissued_at":"2026-05-18T01:34:07.087986Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:07.087986Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.07637","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:34:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DcMLUtz3b76sbkc0e7+jhu0hZcX4PFIzOlUsAQoqwSni1dNg51qTR+6J9DEKZoQVRSEgp6I2YZVag5Ps7o16CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T10:51:30.465883Z"},"content_sha256":"324c118f85a0775c642b39beceb7428bc7e009434d6c544d0b593fccfdc95e4a","schema_version":"1.0","event_id":"sha256:324c118f85a0775c642b39beceb7428bc7e009434d6c544d0b593fccfdc95e4a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:4PWYEDCBZXRVXSPXSIMHUXJQWI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Explicit Expressions for the Variance and Higher Moments of the Size of a Simultaneous Core Partition and its Limiting Distribution","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Doron Zeilberger, Shalosh B. Ekhad","submitted_at":"2015-08-30T21:04:47Z","abstract_excerpt":"Jaclyn Anderson proved that if s and t are relatively prime positive integers, then there are exactly (s+t-1)!/(s!t!) partitions whose set of hook-lengths is disjoint from the set {s,t}. Drew Armstrong conjectured (and Paul Johnson, and a bit later, Victor Wang, proved) a beautiful expression for the average size, namely (s-1)(t-1)(s+t+1)/24 .\n  In the present article, we go far beyond the average, and state absolutely certain expressions (but \"officially\" still conjectures) for the variance (showing in particular that it is rather large, and there is no \"concentration about the mean\"), and th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.07637","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:34:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wRWQ+SsHf44+3lTxpSO1Oh/Wgu7Acrd6w7s2iutNij4CJES8df6Z37k5kKYImrku+opTWzjt0/qQGmEVzMC5AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T10:51:30.466602Z"},"content_sha256":"d79b6401ff48a12a218b7c77f5bb419eab93738ef98c3c420563f43a50807e54","schema_version":"1.0","event_id":"sha256:d79b6401ff48a12a218b7c77f5bb419eab93738ef98c3c420563f43a50807e54"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4PWYEDCBZXRVXSPXSIMHUXJQWI/bundle.json","state_url":"https://pith.science/pith/4PWYEDCBZXRVXSPXSIMHUXJQWI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4PWYEDCBZXRVXSPXSIMHUXJQWI/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-27T10:51:30Z","links":{"resolver":"https://pith.science/pith/4PWYEDCBZXRVXSPXSIMHUXJQWI","bundle":"https://pith.science/pith/4PWYEDCBZXRVXSPXSIMHUXJQWI/bundle.json","state":"https://pith.science/pith/4PWYEDCBZXRVXSPXSIMHUXJQWI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4PWYEDCBZXRVXSPXSIMHUXJQWI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:4PWYEDCBZXRVXSPXSIMHUXJQWI","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"1af2faee046793c9facf254d6c47989000339ea84123e61404a08cc6099dd44d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-30T21:04:47Z","title_canon_sha256":"c09d90ede7c8172d43670024a1f447feabcc22a493811baa8730a756bc1308c6"},"schema_version":"1.0","source":{"id":"1508.07637","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.07637","created_at":"2026-05-18T01:34:07Z"},{"alias_kind":"arxiv_version","alias_value":"1508.07637v2","created_at":"2026-05-18T01:34:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.07637","created_at":"2026-05-18T01:34:07Z"},{"alias_kind":"pith_short_12","alias_value":"4PWYEDCBZXRV","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"4PWYEDCBZXRVXSPX","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"4PWYEDCB","created_at":"2026-05-18T12:29:05Z"}],"graph_snapshots":[{"event_id":"sha256:d79b6401ff48a12a218b7c77f5bb419eab93738ef98c3c420563f43a50807e54","target":"graph","created_at":"2026-05-18T01:34:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Jaclyn Anderson proved that if s and t are relatively prime positive integers, then there are exactly (s+t-1)!/(s!t!) partitions whose set of hook-lengths is disjoint from the set {s,t}. Drew Armstrong conjectured (and Paul Johnson, and a bit later, Victor Wang, proved) a beautiful expression for the average size, namely (s-1)(t-1)(s+t+1)/24 .\n  In the present article, we go far beyond the average, and state absolutely certain expressions (but \"officially\" still conjectures) for the variance (showing in particular that it is rather large, and there is no \"concentration about the mean\"), and th","authors_text":"Doron Zeilberger, Shalosh B. Ekhad","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-30T21:04:47Z","title":"Explicit Expressions for the Variance and Higher Moments of the Size of a Simultaneous Core Partition and its Limiting Distribution"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.07637","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:324c118f85a0775c642b39beceb7428bc7e009434d6c544d0b593fccfdc95e4a","target":"record","created_at":"2026-05-18T01:34:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"1af2faee046793c9facf254d6c47989000339ea84123e61404a08cc6099dd44d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-30T21:04:47Z","title_canon_sha256":"c09d90ede7c8172d43670024a1f447feabcc22a493811baa8730a756bc1308c6"},"schema_version":"1.0","source":{"id":"1508.07637","kind":"arxiv","version":2}},"canonical_sha256":"e3ed820c41cde35bc9f792187a5d30b213c458b5d352763517fec613b6a66b42","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e3ed820c41cde35bc9f792187a5d30b213c458b5d352763517fec613b6a66b42","first_computed_at":"2026-05-18T01:34:07.087986Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:07.087986Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"w2Cv/gXaHm79rM5mvNARK+ZgW1J0Fj0MI9YlZglD0HNLxnHZ7q94BhQTxWNVQnNm8vND5yjmnznnk1atL4RvBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:07.088489Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.07637","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:324c118f85a0775c642b39beceb7428bc7e009434d6c544d0b593fccfdc95e4a","sha256:d79b6401ff48a12a218b7c77f5bb419eab93738ef98c3c420563f43a50807e54"],"state_sha256":"ffc6a11a880bf054717bfa3b04e7c7d981a6b99c0ba8fd6546da92086aa4a164"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"50+ny6nllQ5TFoU6x7/8DjNJN/zOeRYESH1/VzCtssNFGWde5SIJcph7XlHXcTd6kyo2yJNQsDd17Yr5aVhbCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T10:51:30.470263Z","bundle_sha256":"e92ec81298093b65ab372e025117e5040b5f9f78af90af4c84298d857d077283"}}