{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:SHBYR7ILNBAFOIDEIZNAZRWXOS","short_pith_number":"pith:SHBYR7IL","canonical_record":{"source":{"id":"1907.06363","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-15T08:27:14Z","cross_cats_sorted":[],"title_canon_sha256":"f4bc4075b6c2ecf4ef0ca4f99a124dea71e917db2b1a275fdd348618b171b017","abstract_canon_sha256":"febc4dfb3c55cda12129e9d1200af0e7803cf5d340ea1e4646c72333d94dfaea"},"schema_version":"1.0"},"canonical_sha256":"91c388fd0b6840572064465a0cc6d774bd53f8bc00ea87df3053245579980769","source":{"kind":"arxiv","id":"1907.06363","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.06363","created_at":"2026-05-17T23:40:37Z"},{"alias_kind":"arxiv_version","alias_value":"1907.06363v1","created_at":"2026-05-17T23:40:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.06363","created_at":"2026-05-17T23:40:37Z"},{"alias_kind":"pith_short_12","alias_value":"SHBYR7ILNBAF","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"SHBYR7ILNBAFOIDE","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"SHBYR7IL","created_at":"2026-05-18T12:33:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:SHBYR7ILNBAFOIDEIZNAZRWXOS","target":"record","payload":{"canonical_record":{"source":{"id":"1907.06363","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-15T08:27:14Z","cross_cats_sorted":[],"title_canon_sha256":"f4bc4075b6c2ecf4ef0ca4f99a124dea71e917db2b1a275fdd348618b171b017","abstract_canon_sha256":"febc4dfb3c55cda12129e9d1200af0e7803cf5d340ea1e4646c72333d94dfaea"},"schema_version":"1.0"},"canonical_sha256":"91c388fd0b6840572064465a0cc6d774bd53f8bc00ea87df3053245579980769","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:37.817170Z","signature_b64":"rVgobRY5SPsdQaEcpdPjR6LGdU018dzA+AuBRhT+GTjmaNDCcs2RTp/DhKUkLkynOeNh7pdhqEDUSzEcM4uFCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"91c388fd0b6840572064465a0cc6d774bd53f8bc00ea87df3053245579980769","last_reissued_at":"2026-05-17T23:40:37.816512Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:37.816512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1907.06363","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":"2026-05-17T23:40:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gKivv/kMfQVwpOTzck6qLfxTCmCTIwPQRlHcwSVxgWX1cSGc0+NKTMsdW6ozHes0N0c/KYKp1ZVMilFA9wI7CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T04:35:46.804081Z"},"content_sha256":"8ceac3bb7cff30165db074bd85ca97683eb16240eff3cca6706ddfbb4c6a8833","schema_version":"1.0","event_id":"sha256:8ceac3bb7cff30165db074bd85ca97683eb16240eff3cca6706ddfbb4c6a8833"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:SHBYR7ILNBAFOIDEIZNAZRWXOS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Linked partition ideals, directed graphs and $q$-multi-summations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Shane Chern","submitted_at":"2019-07-15T08:27:14Z","abstract_excerpt":"Finding an Andrews--Gordon type generating function identity for a linked partition ideal is difficult in most cases. In this paper, we will handle this problem in the setting of graph theory. With the generating function of directed graphs with an ``empty'' vertex, we then turn our attention to a $q$-difference system. This $q$-difference system eventually yields a factorization problem of a special type of column functional vectors involving $q$-multi-summations. Finally, using a recurrence relation satisfied by certain $q$-multi-summations, we are able to provide non-computer-assisted proof"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.06363","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":"2026-05-17T23:40:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jWU8RCRBmsbRNi6bSXxcv+XpUdX3DBCt3/jzaNrSQ/0ZNaW42xLJsKUB2YrYLbw/G0wV8LyZMCRIIAJdxK5/Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T04:35:46.804462Z"},"content_sha256":"dea9a7d6aa05e91511bf1f0bdf6c93d90e859b917b7ccabe11ceb3d96ae6a4a3","schema_version":"1.0","event_id":"sha256:dea9a7d6aa05e91511bf1f0bdf6c93d90e859b917b7ccabe11ceb3d96ae6a4a3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SHBYR7ILNBAFOIDEIZNAZRWXOS/bundle.json","state_url":"https://pith.science/pith/SHBYR7ILNBAFOIDEIZNAZRWXOS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SHBYR7ILNBAFOIDEIZNAZRWXOS/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":"2026-06-03T04:35:46Z","links":{"resolver":"https://pith.science/pith/SHBYR7ILNBAFOIDEIZNAZRWXOS","bundle":"https://pith.science/pith/SHBYR7ILNBAFOIDEIZNAZRWXOS/bundle.json","state":"https://pith.science/pith/SHBYR7ILNBAFOIDEIZNAZRWXOS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SHBYR7ILNBAFOIDEIZNAZRWXOS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:SHBYR7ILNBAFOIDEIZNAZRWXOS","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":"febc4dfb3c55cda12129e9d1200af0e7803cf5d340ea1e4646c72333d94dfaea","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-15T08:27:14Z","title_canon_sha256":"f4bc4075b6c2ecf4ef0ca4f99a124dea71e917db2b1a275fdd348618b171b017"},"schema_version":"1.0","source":{"id":"1907.06363","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.06363","created_at":"2026-05-17T23:40:37Z"},{"alias_kind":"arxiv_version","alias_value":"1907.06363v1","created_at":"2026-05-17T23:40:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.06363","created_at":"2026-05-17T23:40:37Z"},{"alias_kind":"pith_short_12","alias_value":"SHBYR7ILNBAF","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"SHBYR7ILNBAFOIDE","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"SHBYR7IL","created_at":"2026-05-18T12:33:27Z"}],"graph_snapshots":[{"event_id":"sha256:dea9a7d6aa05e91511bf1f0bdf6c93d90e859b917b7ccabe11ceb3d96ae6a4a3","target":"graph","created_at":"2026-05-17T23:40:37Z","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":"Finding an Andrews--Gordon type generating function identity for a linked partition ideal is difficult in most cases. In this paper, we will handle this problem in the setting of graph theory. With the generating function of directed graphs with an ``empty'' vertex, we then turn our attention to a $q$-difference system. This $q$-difference system eventually yields a factorization problem of a special type of column functional vectors involving $q$-multi-summations. Finally, using a recurrence relation satisfied by certain $q$-multi-summations, we are able to provide non-computer-assisted proof","authors_text":"Shane Chern","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-15T08:27:14Z","title":"Linked partition ideals, directed graphs and $q$-multi-summations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.06363","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":"sha256:8ceac3bb7cff30165db074bd85ca97683eb16240eff3cca6706ddfbb4c6a8833","target":"record","created_at":"2026-05-17T23:40:37Z","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":"febc4dfb3c55cda12129e9d1200af0e7803cf5d340ea1e4646c72333d94dfaea","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-15T08:27:14Z","title_canon_sha256":"f4bc4075b6c2ecf4ef0ca4f99a124dea71e917db2b1a275fdd348618b171b017"},"schema_version":"1.0","source":{"id":"1907.06363","kind":"arxiv","version":1}},"canonical_sha256":"91c388fd0b6840572064465a0cc6d774bd53f8bc00ea87df3053245579980769","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"91c388fd0b6840572064465a0cc6d774bd53f8bc00ea87df3053245579980769","first_computed_at":"2026-05-17T23:40:37.816512Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:40:37.816512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rVgobRY5SPsdQaEcpdPjR6LGdU018dzA+AuBRhT+GTjmaNDCcs2RTp/DhKUkLkynOeNh7pdhqEDUSzEcM4uFCQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:40:37.817170Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.06363","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8ceac3bb7cff30165db074bd85ca97683eb16240eff3cca6706ddfbb4c6a8833","sha256:dea9a7d6aa05e91511bf1f0bdf6c93d90e859b917b7ccabe11ceb3d96ae6a4a3"],"state_sha256":"6996bf1f2963af5f41187fa9bf1ec3ede5e617bfc88aca535a91e81066cd894d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MWF3Bc5wmmIiLqPs0qlDJdxAmtihUI30IL65EoBxAiNdLH2Hitq0b0M7tKhffE6H4xfuO+Li/jW06OQWEcm9CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T04:35:46.806798Z","bundle_sha256":"12eb8ba77f9f5b0377f905fc85ccb014da907a72b87e01d296adbcd2c62b6677"}}