{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:LOSRBZ6YTABOKN6HS7VJXGT7IX","short_pith_number":"pith:LOSRBZ6Y","canonical_record":{"source":{"id":"1109.1236","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-06T17:26:55Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"42ecc56da55d29ebda20d4c60920752676af5f50ec5899faf31bc6fb2896f790","abstract_canon_sha256":"fc0fec2dcf848a8021f56ef07256a051422501762b84209be693a860888b8556"},"schema_version":"1.0"},"canonical_sha256":"5ba510e7d89802e537c797ea9b9a7f45ffc2733ae21b899aff1c35df88f26cb1","source":{"kind":"arxiv","id":"1109.1236","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.1236","created_at":"2026-05-18T03:43:18Z"},{"alias_kind":"arxiv_version","alias_value":"1109.1236v2","created_at":"2026-05-18T03:43:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.1236","created_at":"2026-05-18T03:43:18Z"},{"alias_kind":"pith_short_12","alias_value":"LOSRBZ6YTABO","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"LOSRBZ6YTABOKN6H","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"LOSRBZ6Y","created_at":"2026-05-18T12:26:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:LOSRBZ6YTABOKN6HS7VJXGT7IX","target":"record","payload":{"canonical_record":{"source":{"id":"1109.1236","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-06T17:26:55Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"42ecc56da55d29ebda20d4c60920752676af5f50ec5899faf31bc6fb2896f790","abstract_canon_sha256":"fc0fec2dcf848a8021f56ef07256a051422501762b84209be693a860888b8556"},"schema_version":"1.0"},"canonical_sha256":"5ba510e7d89802e537c797ea9b9a7f45ffc2733ae21b899aff1c35df88f26cb1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:43:18.734376Z","signature_b64":"YM6umGz4Q+qAJL+jAPyDM3s2sndXOsW/HsM/J/sW/Lk9VN1ZRmVMgVmttvHuJEyCT7I1tzVvNv4y2USy1njPBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ba510e7d89802e537c797ea9b9a7f45ffc2733ae21b899aff1c35df88f26cb1","last_reissued_at":"2026-05-18T03:43:18.733559Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:43:18.733559Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.1236","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":"2026-05-18T03:43:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zUxsetglWUGkjESJES1qcGFCsFD97GC5SHRkGFoTJZgaV8RRQtWRIDK8IaAmMc0dehmvhWPG48p0oJ8h2fSDDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T19:02:52.455763Z"},"content_sha256":"d4e9578254b08d762a1823b98029bce3aa85912a89df5f98bfb9c0cb19865e48","schema_version":"1.0","event_id":"sha256:d4e9578254b08d762a1823b98029bce3aa85912a89df5f98bfb9c0cb19865e48"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:LOSRBZ6YTABOKN6HS7VJXGT7IX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Polynomial analogues of Ramanujan congruences for Han's hooklength formula","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"William J. Keith","submitted_at":"2011-09-06T17:26:55Z","abstract_excerpt":"This article considers the eta power $\\prod {(1-q^k)}^{b-1}$. It is proved that the coefficients of $\\frac{q^n}{n!}$ in this expression, as polynomials in $b$, exhibit equidistribution of the coefficients in the nonzero residue classes mod 5 when $n=5j+4$. Other symmetries, as well as symmetries for other primes and prime powers, are proved, and some open questions are raised."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.1236","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":"2026-05-18T03:43:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oS2huvP8DfsU1Q3NE/VnYpD4fq+NEQUTk9gkg9INJ/iGM3xj1Uhq3wEqdkkBokWwZ3SaYPFwr6u7eZZ3hVKzBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T19:02:52.456148Z"},"content_sha256":"c327669f2d2a0c46b5bcfdcf48e9c1f21cd5c263154fec5bc018543de7122869","schema_version":"1.0","event_id":"sha256:c327669f2d2a0c46b5bcfdcf48e9c1f21cd5c263154fec5bc018543de7122869"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LOSRBZ6YTABOKN6HS7VJXGT7IX/bundle.json","state_url":"https://pith.science/pith/LOSRBZ6YTABOKN6HS7VJXGT7IX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LOSRBZ6YTABOKN6HS7VJXGT7IX/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":"2026-06-06T19:02:52Z","links":{"resolver":"https://pith.science/pith/LOSRBZ6YTABOKN6HS7VJXGT7IX","bundle":"https://pith.science/pith/LOSRBZ6YTABOKN6HS7VJXGT7IX/bundle.json","state":"https://pith.science/pith/LOSRBZ6YTABOKN6HS7VJXGT7IX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LOSRBZ6YTABOKN6HS7VJXGT7IX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:LOSRBZ6YTABOKN6HS7VJXGT7IX","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":"fc0fec2dcf848a8021f56ef07256a051422501762b84209be693a860888b8556","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-06T17:26:55Z","title_canon_sha256":"42ecc56da55d29ebda20d4c60920752676af5f50ec5899faf31bc6fb2896f790"},"schema_version":"1.0","source":{"id":"1109.1236","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.1236","created_at":"2026-05-18T03:43:18Z"},{"alias_kind":"arxiv_version","alias_value":"1109.1236v2","created_at":"2026-05-18T03:43:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.1236","created_at":"2026-05-18T03:43:18Z"},{"alias_kind":"pith_short_12","alias_value":"LOSRBZ6YTABO","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"LOSRBZ6YTABOKN6H","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"LOSRBZ6Y","created_at":"2026-05-18T12:26:34Z"}],"graph_snapshots":[{"event_id":"sha256:c327669f2d2a0c46b5bcfdcf48e9c1f21cd5c263154fec5bc018543de7122869","target":"graph","created_at":"2026-05-18T03:43:18Z","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":"This article considers the eta power $\\prod {(1-q^k)}^{b-1}$. It is proved that the coefficients of $\\frac{q^n}{n!}$ in this expression, as polynomials in $b$, exhibit equidistribution of the coefficients in the nonzero residue classes mod 5 when $n=5j+4$. Other symmetries, as well as symmetries for other primes and prime powers, are proved, and some open questions are raised.","authors_text":"William J. Keith","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-06T17:26:55Z","title":"Polynomial analogues of Ramanujan congruences for Han's hooklength formula"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.1236","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:d4e9578254b08d762a1823b98029bce3aa85912a89df5f98bfb9c0cb19865e48","target":"record","created_at":"2026-05-18T03:43:18Z","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":"fc0fec2dcf848a8021f56ef07256a051422501762b84209be693a860888b8556","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-06T17:26:55Z","title_canon_sha256":"42ecc56da55d29ebda20d4c60920752676af5f50ec5899faf31bc6fb2896f790"},"schema_version":"1.0","source":{"id":"1109.1236","kind":"arxiv","version":2}},"canonical_sha256":"5ba510e7d89802e537c797ea9b9a7f45ffc2733ae21b899aff1c35df88f26cb1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5ba510e7d89802e537c797ea9b9a7f45ffc2733ae21b899aff1c35df88f26cb1","first_computed_at":"2026-05-18T03:43:18.733559Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:43:18.733559Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YM6umGz4Q+qAJL+jAPyDM3s2sndXOsW/HsM/J/sW/Lk9VN1ZRmVMgVmttvHuJEyCT7I1tzVvNv4y2USy1njPBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:43:18.734376Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.1236","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d4e9578254b08d762a1823b98029bce3aa85912a89df5f98bfb9c0cb19865e48","sha256:c327669f2d2a0c46b5bcfdcf48e9c1f21cd5c263154fec5bc018543de7122869"],"state_sha256":"8762188a016bbf33f48d542545ff58d050272ecdcd1f90e49ddac93e86eeecc8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lJ2lg0TyZXgzOf8caJ9rn4zkapliUB0uL8QWHnE/XwEWHKdlOcfat058SDKw1m9+bS5uuq0dXf+zoXeYhQWoCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T19:02:52.458219Z","bundle_sha256":"be86a5f107e9999d2eb980882413a6dd48ecd8b675f12ab11e095e6835708ee4"}}