{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:ZCTS33WCBSVO6F2RPJTC766AYI","short_pith_number":"pith:ZCTS33WC","canonical_record":{"source":{"id":"1805.10998","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-28T16:01:16Z","cross_cats_sorted":[],"title_canon_sha256":"f4d0d478b7756a960a71183f1587e8196292e6d2a33a1e5f4f5e46aa64fc1cc5","abstract_canon_sha256":"7d02c4378586eb96d44a9efda359c1c2d3ad26d6af0a2ff67a7776688e0e6554"},"schema_version":"1.0"},"canonical_sha256":"c8a72deec20caaef17517a662ffbc0c203c2a482fd3f09459e6275bbd9f0f5ff","source":{"kind":"arxiv","id":"1805.10998","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.10998","created_at":"2026-05-18T00:14:48Z"},{"alias_kind":"arxiv_version","alias_value":"1805.10998v1","created_at":"2026-05-18T00:14:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.10998","created_at":"2026-05-18T00:14:48Z"},{"alias_kind":"pith_short_12","alias_value":"ZCTS33WCBSVO","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"ZCTS33WCBSVO6F2R","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"ZCTS33WC","created_at":"2026-05-18T12:33:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:ZCTS33WCBSVO6F2RPJTC766AYI","target":"record","payload":{"canonical_record":{"source":{"id":"1805.10998","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-28T16:01:16Z","cross_cats_sorted":[],"title_canon_sha256":"f4d0d478b7756a960a71183f1587e8196292e6d2a33a1e5f4f5e46aa64fc1cc5","abstract_canon_sha256":"7d02c4378586eb96d44a9efda359c1c2d3ad26d6af0a2ff67a7776688e0e6554"},"schema_version":"1.0"},"canonical_sha256":"c8a72deec20caaef17517a662ffbc0c203c2a482fd3f09459e6275bbd9f0f5ff","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:48.557683Z","signature_b64":"gwuC5X9W+S3IAIYRSUaBLZ4HswByA7SUHn1xO9MCV8OGVtfjpb8SyQ7mX6JZo+f+vdX0X4FPOtLU30j00jnNDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c8a72deec20caaef17517a662ffbc0c203c2a482fd3f09459e6275bbd9f0f5ff","last_reissued_at":"2026-05-18T00:14:48.556987Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:48.556987Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.10998","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:14:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"10iLW6ct5ADGeSu659EP28vlPokZ5QuQL9j6FizcjT5z3TutWLoArXFDC9KIE9z5GeBhmshy8g3oC/dgeaegDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T17:01:42.433306Z"},"content_sha256":"611e661ce5bbd1786203de9bc64793d679b92af70b422e410520c1eba6279b92","schema_version":"1.0","event_id":"sha256:611e661ce5bbd1786203de9bc64793d679b92af70b422e410520c1eba6279b92"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:ZCTS33WCBSVO6F2RPJTC766AYI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On certain combinatorial expansions of the Legendre-Stirling numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jun Ma, Shi-Mei Ma, Yeong-Nan Yeh","submitted_at":"2018-05-28T16:01:16Z","abstract_excerpt":"The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. In this paper, we provide a combinatorial code for Legendre-Stirling set partitions. As an application, we obtain combinatorial expansions of the Legendre-Stirling numbers of both kinds. Moreover, we present grammatical descriptions of the Jacobi-Stirling numbers of both kinds."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.10998","kind":"arxiv","version":1},"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":"2026-05-18T00:14:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"G5V6ypeggIC8MnIh4qYbY/OD5kBSgWjZvM7YsRUtCKfoWuDSbqzTTzpG+U/yyhI2oOJ6epqVEXqt5VgG+ndDBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T17:01:42.433666Z"},"content_sha256":"e886e0437624986b2b0e62f017128e4c05bc48cba607dccb1980e5c2cbd9aaa9","schema_version":"1.0","event_id":"sha256:e886e0437624986b2b0e62f017128e4c05bc48cba607dccb1980e5c2cbd9aaa9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZCTS33WCBSVO6F2RPJTC766AYI/bundle.json","state_url":"https://pith.science/pith/ZCTS33WCBSVO6F2RPJTC766AYI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZCTS33WCBSVO6F2RPJTC766AYI/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":"2026-06-01T17:01:42Z","links":{"resolver":"https://pith.science/pith/ZCTS33WCBSVO6F2RPJTC766AYI","bundle":"https://pith.science/pith/ZCTS33WCBSVO6F2RPJTC766AYI/bundle.json","state":"https://pith.science/pith/ZCTS33WCBSVO6F2RPJTC766AYI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZCTS33WCBSVO6F2RPJTC766AYI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:ZCTS33WCBSVO6F2RPJTC766AYI","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":"7d02c4378586eb96d44a9efda359c1c2d3ad26d6af0a2ff67a7776688e0e6554","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-28T16:01:16Z","title_canon_sha256":"f4d0d478b7756a960a71183f1587e8196292e6d2a33a1e5f4f5e46aa64fc1cc5"},"schema_version":"1.0","source":{"id":"1805.10998","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.10998","created_at":"2026-05-18T00:14:48Z"},{"alias_kind":"arxiv_version","alias_value":"1805.10998v1","created_at":"2026-05-18T00:14:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.10998","created_at":"2026-05-18T00:14:48Z"},{"alias_kind":"pith_short_12","alias_value":"ZCTS33WCBSVO","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"ZCTS33WCBSVO6F2R","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"ZCTS33WC","created_at":"2026-05-18T12:33:04Z"}],"graph_snapshots":[{"event_id":"sha256:e886e0437624986b2b0e62f017128e4c05bc48cba607dccb1980e5c2cbd9aaa9","target":"graph","created_at":"2026-05-18T00:14:48Z","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":"The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. In this paper, we provide a combinatorial code for Legendre-Stirling set partitions. As an application, we obtain combinatorial expansions of the Legendre-Stirling numbers of both kinds. Moreover, we present grammatical descriptions of the Jacobi-Stirling numbers of both kinds.","authors_text":"Jun Ma, Shi-Mei Ma, Yeong-Nan Yeh","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-28T16:01:16Z","title":"On certain combinatorial expansions of the Legendre-Stirling numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.10998","kind":"arxiv","version":1},"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":"sha256:611e661ce5bbd1786203de9bc64793d679b92af70b422e410520c1eba6279b92","target":"record","created_at":"2026-05-18T00:14:48Z","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":"7d02c4378586eb96d44a9efda359c1c2d3ad26d6af0a2ff67a7776688e0e6554","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-28T16:01:16Z","title_canon_sha256":"f4d0d478b7756a960a71183f1587e8196292e6d2a33a1e5f4f5e46aa64fc1cc5"},"schema_version":"1.0","source":{"id":"1805.10998","kind":"arxiv","version":1}},"canonical_sha256":"c8a72deec20caaef17517a662ffbc0c203c2a482fd3f09459e6275bbd9f0f5ff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c8a72deec20caaef17517a662ffbc0c203c2a482fd3f09459e6275bbd9f0f5ff","first_computed_at":"2026-05-18T00:14:48.556987Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:48.556987Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gwuC5X9W+S3IAIYRSUaBLZ4HswByA7SUHn1xO9MCV8OGVtfjpb8SyQ7mX6JZo+f+vdX0X4FPOtLU30j00jnNDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:48.557683Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.10998","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:611e661ce5bbd1786203de9bc64793d679b92af70b422e410520c1eba6279b92","sha256:e886e0437624986b2b0e62f017128e4c05bc48cba607dccb1980e5c2cbd9aaa9"],"state_sha256":"e622484d860770c5c60b0dab9bf6fa60bfa5cf2c5c064c5418334108e0d2744d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VER7YUFET6gLYp37+qOuX4mmUImAS9IOqty8ILMhmN62LRoErzR86/8XEsOXS5wt70byZZUi+CTy0Lti4BWrCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T17:01:42.435634Z","bundle_sha256":"d61d9ec41a798a2f9f3eaa08988ff618d3b9a404265a0cd845d0cdf3d61d6773"}}