{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:LR4DCJUCSWFSAAZGBKBQHGEXOY","short_pith_number":"pith:LR4DCJUC","canonical_record":{"source":{"id":"0906.0487","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-06-02T14:11:37Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"e67c1e3982fe003195a5ea04be8964d2121c7e101d39929c87f2b6bcbc33dc19","abstract_canon_sha256":"c2427cec648c1f2fed1c74236be4535b98a01cc06a690f48fe094a851c2c54de"},"schema_version":"1.0"},"canonical_sha256":"5c78312682958b2003260a830398977604323eb1ccb4a1e95a6449758b529d4b","source":{"kind":"arxiv","id":"0906.0487","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.0487","created_at":"2026-05-18T04:23:21Z"},{"alias_kind":"arxiv_version","alias_value":"0906.0487v3","created_at":"2026-05-18T04:23:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.0487","created_at":"2026-05-18T04:23:21Z"},{"alias_kind":"pith_short_12","alias_value":"LR4DCJUCSWFS","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"LR4DCJUCSWFSAAZG","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"LR4DCJUC","created_at":"2026-05-18T12:26:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:LR4DCJUCSWFSAAZGBKBQHGEXOY","target":"record","payload":{"canonical_record":{"source":{"id":"0906.0487","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-06-02T14:11:37Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"e67c1e3982fe003195a5ea04be8964d2121c7e101d39929c87f2b6bcbc33dc19","abstract_canon_sha256":"c2427cec648c1f2fed1c74236be4535b98a01cc06a690f48fe094a851c2c54de"},"schema_version":"1.0"},"canonical_sha256":"5c78312682958b2003260a830398977604323eb1ccb4a1e95a6449758b529d4b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:21.306165Z","signature_b64":"/3fTWaN9M26cywQXOwnZKBC6dNiYyAcjtvBabpXN2PYJEEKidrEZGm3Ld50a8R7VMmQBbktFhUUrfKlmUXpACw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c78312682958b2003260a830398977604323eb1ccb4a1e95a6449758b529d4b","last_reissued_at":"2026-05-18T04:23:21.305611Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:21.305611Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0906.0487","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:23:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l7gEpci7/AqItG+FP3/Xb5w7pDaUgAyYoPQs34HG/86Y3V2I64uGbXFyjBpxez6M0okjkZOalWwwuuZd+w7uCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T03:43:51.400798Z"},"content_sha256":"5994fed4f3ba071a4325a2731d8ce60479548ddb4fb9cf8e8bb351f315b05a32","schema_version":"1.0","event_id":"sha256:5994fed4f3ba071a4325a2731d8ce60479548ddb4fb9cf8e8bb351f315b05a32"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:LR4DCJUCSWFSAAZGBKBQHGEXOY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counting the number of elements in the mutation classes of \\tilde{A}_n-quivers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CO","authors_text":"Christian Stump, Janine Bastian, Martin Rubey, Thomas Prellberg","submitted_at":"2009-06-02T14:11:37Z","abstract_excerpt":"In this article we prove explicit formulae for the number of non-isomorphic cluster-tilted algebras of type \\tilde{A}_n in the derived equivalence classes. In particular, we obtain the number of elements in the mutation classes of quivers of type \\tilde{A}_n. As a by-product, this provides an alternative proof for the number of quivers of Dynkin type D_n which was first determined by Buan and Torkildsen."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.0487","kind":"arxiv","version":3},"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":"2026-05-18T04:23:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sjVgRYT/Xgei2AWKRIQL8Yqh+GBcbhPYgXcV4Q5w2MjVAm8oB9keNXo8UaEhVjP3t/iTCptI0cIQvio+6z/DDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T03:43:51.401130Z"},"content_sha256":"9acd11c1a8626683cf89b202c4315e85759029679db66e8236805604fff4557b","schema_version":"1.0","event_id":"sha256:9acd11c1a8626683cf89b202c4315e85759029679db66e8236805604fff4557b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LR4DCJUCSWFSAAZGBKBQHGEXOY/bundle.json","state_url":"https://pith.science/pith/LR4DCJUCSWFSAAZGBKBQHGEXOY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LR4DCJUCSWFSAAZGBKBQHGEXOY/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":"2026-06-20T03:43:51Z","links":{"resolver":"https://pith.science/pith/LR4DCJUCSWFSAAZGBKBQHGEXOY","bundle":"https://pith.science/pith/LR4DCJUCSWFSAAZGBKBQHGEXOY/bundle.json","state":"https://pith.science/pith/LR4DCJUCSWFSAAZGBKBQHGEXOY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LR4DCJUCSWFSAAZGBKBQHGEXOY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:LR4DCJUCSWFSAAZGBKBQHGEXOY","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":"c2427cec648c1f2fed1c74236be4535b98a01cc06a690f48fe094a851c2c54de","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-06-02T14:11:37Z","title_canon_sha256":"e67c1e3982fe003195a5ea04be8964d2121c7e101d39929c87f2b6bcbc33dc19"},"schema_version":"1.0","source":{"id":"0906.0487","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.0487","created_at":"2026-05-18T04:23:21Z"},{"alias_kind":"arxiv_version","alias_value":"0906.0487v3","created_at":"2026-05-18T04:23:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.0487","created_at":"2026-05-18T04:23:21Z"},{"alias_kind":"pith_short_12","alias_value":"LR4DCJUCSWFS","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"LR4DCJUCSWFSAAZG","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"LR4DCJUC","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:9acd11c1a8626683cf89b202c4315e85759029679db66e8236805604fff4557b","target":"graph","created_at":"2026-05-18T04:23:21Z","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":"In this article we prove explicit formulae for the number of non-isomorphic cluster-tilted algebras of type \\tilde{A}_n in the derived equivalence classes. In particular, we obtain the number of elements in the mutation classes of quivers of type \\tilde{A}_n. As a by-product, this provides an alternative proof for the number of quivers of Dynkin type D_n which was first determined by Buan and Torkildsen.","authors_text":"Christian Stump, Janine Bastian, Martin Rubey, Thomas Prellberg","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-06-02T14:11:37Z","title":"Counting the number of elements in the mutation classes of \\tilde{A}_n-quivers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.0487","kind":"arxiv","version":3},"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":"sha256:5994fed4f3ba071a4325a2731d8ce60479548ddb4fb9cf8e8bb351f315b05a32","target":"record","created_at":"2026-05-18T04:23:21Z","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":"c2427cec648c1f2fed1c74236be4535b98a01cc06a690f48fe094a851c2c54de","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-06-02T14:11:37Z","title_canon_sha256":"e67c1e3982fe003195a5ea04be8964d2121c7e101d39929c87f2b6bcbc33dc19"},"schema_version":"1.0","source":{"id":"0906.0487","kind":"arxiv","version":3}},"canonical_sha256":"5c78312682958b2003260a830398977604323eb1ccb4a1e95a6449758b529d4b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5c78312682958b2003260a830398977604323eb1ccb4a1e95a6449758b529d4b","first_computed_at":"2026-05-18T04:23:21.305611Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:23:21.305611Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/3fTWaN9M26cywQXOwnZKBC6dNiYyAcjtvBabpXN2PYJEEKidrEZGm3Ld50a8R7VMmQBbktFhUUrfKlmUXpACw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:23:21.306165Z","signed_message":"canonical_sha256_bytes"},"source_id":"0906.0487","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5994fed4f3ba071a4325a2731d8ce60479548ddb4fb9cf8e8bb351f315b05a32","sha256:9acd11c1a8626683cf89b202c4315e85759029679db66e8236805604fff4557b"],"state_sha256":"a26bea23ee139dba77ce7f0d5f7f490ff367ea81ee4469f1bef50a91616120ca"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ISfNkVOPfNWfyRKX+NJPamap6ET/ctDdIYrZl124VGlu1Eg/705SaQU1FD1BaKv0Kg63rlF5NHY1gvk1xhHuCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T03:43:51.403092Z","bundle_sha256":"30b6fd1c21499584ef0a5b26b9aa261bec56f8b8c7ce5d8b0500979e67066c44"}}