{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:EYQC5XUFBEUZYBU36KEHDCQGWT","short_pith_number":"pith:EYQC5XUF","canonical_record":{"source":{"id":"1507.02852","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-07-10T11:15:21Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"445933a6fc20c2f28dbc813f9da94b98b173870e30452fef683fc9e5d3deb51b","abstract_canon_sha256":"1c617f7df3d0c01d8a921dbc9c5eda65201b6a0a15961a6900c8adf3675834fb"},"schema_version":"1.0"},"canonical_sha256":"26202ede8509299c069bf288718a06b4fa8916de8e8264dbe5bafaf817af2cac","source":{"kind":"arxiv","id":"1507.02852","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.02852","created_at":"2026-05-18T01:37:03Z"},{"alias_kind":"arxiv_version","alias_value":"1507.02852v1","created_at":"2026-05-18T01:37:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.02852","created_at":"2026-05-18T01:37:03Z"},{"alias_kind":"pith_short_12","alias_value":"EYQC5XUFBEUZ","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"EYQC5XUFBEUZYBU3","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"EYQC5XUF","created_at":"2026-05-18T12:29:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:EYQC5XUFBEUZYBU36KEHDCQGWT","target":"record","payload":{"canonical_record":{"source":{"id":"1507.02852","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-07-10T11:15:21Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"445933a6fc20c2f28dbc813f9da94b98b173870e30452fef683fc9e5d3deb51b","abstract_canon_sha256":"1c617f7df3d0c01d8a921dbc9c5eda65201b6a0a15961a6900c8adf3675834fb"},"schema_version":"1.0"},"canonical_sha256":"26202ede8509299c069bf288718a06b4fa8916de8e8264dbe5bafaf817af2cac","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:03.064465Z","signature_b64":"23PsLaTGkmUbTR+gL9svkMISwprNdYYJyM+TTimELStxx36YMYnS54oRStlMqHWwbCgEj5t2l3mwHvaSivgCCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"26202ede8509299c069bf288718a06b4fa8916de8e8264dbe5bafaf817af2cac","last_reissued_at":"2026-05-18T01:37:03.063992Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:03.063992Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.02852","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":"2026-05-18T01:37:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"50lZo9/U/eVp6Oqif+buWvz2GzirO40ZUwp6NjTUbkkM4pI0LHeWxmBtVuTzgsgoKzVRvHJYDuSfME7iB3r4BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T22:46:30.492309Z"},"content_sha256":"cb4885a594e880f4ea3a5d36a827c9469dce5ed1ca0d1c6934a1d649a036b68c","schema_version":"1.0","event_id":"sha256:cb4885a594e880f4ea3a5d36a827c9469dce5ed1ca0d1c6934a1d649a036b68c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:EYQC5XUFBEUZYBU36KEHDCQGWT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Remarks on lengths of maximal green sequences for quivers of type $\\tilde{A}_{n,1}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RT","authors_text":"Ryoichi Kase","submitted_at":"2015-07-10T11:15:21Z","abstract_excerpt":"A maximal green sequence introduced by B. Keller is a certain sequence of quiver mutations at green vertices. T. Br\\\"ustle, G. Dupont and M. P\\'erotin showed that for an acyclic quiver, maximal green sequences are realized as maximal paths in the Hasse quiver of the poset of support tilting modules. They also considered possible lengths of maximal green sequences. In this paper, we calculate possible lengths of maximal green sequences for a quiver of type $A$ or of type $\\tilde{A}_{n,1}$ by using theory of tilting mutation."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.02852","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":"2026-05-18T01:37:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"clWgsjkJwYAI1DEYq5cJ3e4CMZ0G/IApa+vlUx7rviSQ5YQWBBJiKqqJPO1vuUEiYNlWiXh2HZjjaRQXBMJbBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T22:46:30.492674Z"},"content_sha256":"d4b254a246d0ca79564761655fc1db3a6d788a96dc365cae31ea81dd63e15ae9","schema_version":"1.0","event_id":"sha256:d4b254a246d0ca79564761655fc1db3a6d788a96dc365cae31ea81dd63e15ae9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EYQC5XUFBEUZYBU36KEHDCQGWT/bundle.json","state_url":"https://pith.science/pith/EYQC5XUFBEUZYBU36KEHDCQGWT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EYQC5XUFBEUZYBU36KEHDCQGWT/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":"2026-06-30T22:46:30Z","links":{"resolver":"https://pith.science/pith/EYQC5XUFBEUZYBU36KEHDCQGWT","bundle":"https://pith.science/pith/EYQC5XUFBEUZYBU36KEHDCQGWT/bundle.json","state":"https://pith.science/pith/EYQC5XUFBEUZYBU36KEHDCQGWT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EYQC5XUFBEUZYBU36KEHDCQGWT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:EYQC5XUFBEUZYBU36KEHDCQGWT","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":"1c617f7df3d0c01d8a921dbc9c5eda65201b6a0a15961a6900c8adf3675834fb","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-07-10T11:15:21Z","title_canon_sha256":"445933a6fc20c2f28dbc813f9da94b98b173870e30452fef683fc9e5d3deb51b"},"schema_version":"1.0","source":{"id":"1507.02852","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.02852","created_at":"2026-05-18T01:37:03Z"},{"alias_kind":"arxiv_version","alias_value":"1507.02852v1","created_at":"2026-05-18T01:37:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.02852","created_at":"2026-05-18T01:37:03Z"},{"alias_kind":"pith_short_12","alias_value":"EYQC5XUFBEUZ","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"EYQC5XUFBEUZYBU3","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"EYQC5XUF","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:d4b254a246d0ca79564761655fc1db3a6d788a96dc365cae31ea81dd63e15ae9","target":"graph","created_at":"2026-05-18T01:37:03Z","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":"A maximal green sequence introduced by B. Keller is a certain sequence of quiver mutations at green vertices. T. Br\\\"ustle, G. Dupont and M. P\\'erotin showed that for an acyclic quiver, maximal green sequences are realized as maximal paths in the Hasse quiver of the poset of support tilting modules. They also considered possible lengths of maximal green sequences. In this paper, we calculate possible lengths of maximal green sequences for a quiver of type $A$ or of type $\\tilde{A}_{n,1}$ by using theory of tilting mutation.","authors_text":"Ryoichi Kase","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-07-10T11:15:21Z","title":"Remarks on lengths of maximal green sequences for quivers of type $\\tilde{A}_{n,1}$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.02852","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:cb4885a594e880f4ea3a5d36a827c9469dce5ed1ca0d1c6934a1d649a036b68c","target":"record","created_at":"2026-05-18T01:37:03Z","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":"1c617f7df3d0c01d8a921dbc9c5eda65201b6a0a15961a6900c8adf3675834fb","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-07-10T11:15:21Z","title_canon_sha256":"445933a6fc20c2f28dbc813f9da94b98b173870e30452fef683fc9e5d3deb51b"},"schema_version":"1.0","source":{"id":"1507.02852","kind":"arxiv","version":1}},"canonical_sha256":"26202ede8509299c069bf288718a06b4fa8916de8e8264dbe5bafaf817af2cac","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"26202ede8509299c069bf288718a06b4fa8916de8e8264dbe5bafaf817af2cac","first_computed_at":"2026-05-18T01:37:03.063992Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:37:03.063992Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"23PsLaTGkmUbTR+gL9svkMISwprNdYYJyM+TTimELStxx36YMYnS54oRStlMqHWwbCgEj5t2l3mwHvaSivgCCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:37:03.064465Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.02852","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cb4885a594e880f4ea3a5d36a827c9469dce5ed1ca0d1c6934a1d649a036b68c","sha256:d4b254a246d0ca79564761655fc1db3a6d788a96dc365cae31ea81dd63e15ae9"],"state_sha256":"52d15731db4f3499b376c8a4d3e547ff9d964352599534446a271e57087cb5f6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"phbwDm9qAhDwZUeDl5cZFAJ1V7T9ksReb9XC8t8VygjrvHyKjUXDw0FzeQCsxF1MmjZoCtIr5Y0LpZxr+3hWDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T22:46:30.494771Z","bundle_sha256":"c1c832214746424cd8fd70626a13672b702bfc4b95c892744ecf5b914e8e1060"}}