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We describe an induced sequence of simple (backward) tilting in the bounded derived category $\\mathcal{D}(Q)$, starting from the standard heart $\\mathcal{H}_Q=\\operatorname{mod}\\mathbf{k}Q$ and ending at another heart $\\mathcal{H}_\\mathbf{s}$ in $\\mathcal{D}(Q)$. Then we show that $\\mathbf{s}$ is a green mutation sequence if and only if every heart in this simple tilting sequence is greater than or equal to $\\mathcal{H}_Q[-1]$; it is maximal if and only if $\\mathcal{H}_\\mathbf{s}=\\mathcal{H}_Q[-1"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1205.0034","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-04-30T21:30:47Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"38f8ecbbdffd3efa87903472719bc146b5e4ae750a88eb2fa1a059b226a79c7c","abstract_canon_sha256":"e74ae80aaf1fb5550fc249677624036171ee87dc3b426bb397af1bc2b7ee3260"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:30.382418Z","signature_b64":"zkfT9Mb2XCTk4AECgAwqAh9eDtDJkt29BjX4xmw9VCrGxljuwwB6Mjb+SksFyrc9AtOOXrwTxRNi6+0h0CSYAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c391a19ba985fe4cb72cb75ced04cfc0e55f757604e9c2149bff726e5dcfda5","last_reissued_at":"2026-05-18T00:44:30.382029Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:30.382029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"C-sortable words as green mutation sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CO","authors_text":"Yu Qiu","submitted_at":"2012-04-30T21:30:47Z","abstract_excerpt":"Let $Q$ be an acyclic quiver and $\\mathbf{s}$ be a sequence with elements in the vertex set $Q_0$. 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Then we show that $\\mathbf{s}$ is a green mutation sequence if and only if every heart in this simple tilting sequence is greater than or equal to $\\mathcal{H}_Q[-1]$; it is maximal if and only if $\\mathcal{H}_\\mathbf{s}=\\mathcal{H}_Q[-1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.0034","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1205.0034","created_at":"2026-05-18T00:44:30.382079+00:00"},{"alias_kind":"arxiv_version","alias_value":"1205.0034v3","created_at":"2026-05-18T00:44:30.382079+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.0034","created_at":"2026-05-18T00:44:30.382079+00:00"},{"alias_kind":"pith_short_12","alias_value":"PQ4RUGN2TBP6","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_16","alias_value":"PQ4RUGN2TBP6JS3S","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_8","alias_value":"PQ4RUGN2","created_at":"2026-05-18T12:27:18.751474+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PQ4RUGN2TBP6JS3SZN245UCM7Q","json":"https://pith.science/pith/PQ4RUGN2TBP6JS3SZN245UCM7Q.json","graph_json":"https://pith.science/api/pith-number/PQ4RUGN2TBP6JS3SZN245UCM7Q/graph.json","events_json":"https://pith.science/api/pith-number/PQ4RUGN2TBP6JS3SZN245UCM7Q/events.json","paper":"https://pith.science/paper/PQ4RUGN2"},"agent_actions":{"view_html":"https://pith.science/pith/PQ4RUGN2TBP6JS3SZN245UCM7Q","download_json":"https://pith.science/pith/PQ4RUGN2TBP6JS3SZN245UCM7Q.json","view_paper":"https://pith.science/paper/PQ4RUGN2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1205.0034&json=true","fetch_graph":"https://pith.science/api/pith-number/PQ4RUGN2TBP6JS3SZN245UCM7Q/graph.json","fetch_events":"https://pith.science/api/pith-number/PQ4RUGN2TBP6JS3SZN245UCM7Q/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PQ4RUGN2TBP6JS3SZN245UCM7Q/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PQ4RUGN2TBP6JS3SZN245UCM7Q/action/storage_attestation","attest_author":"https://pith.science/pith/PQ4RUGN2TBP6JS3SZN245UCM7Q/action/author_attestation","sign_citation":"https://pith.science/pith/PQ4RUGN2TBP6JS3SZN245UCM7Q/action/citation_signature","submit_replication":"https://pith.science/pith/PQ4RUGN2TBP6JS3SZN245UCM7Q/action/replication_record"}},"created_at":"2026-05-18T00:44:30.382079+00:00","updated_at":"2026-05-18T00:44:30.382079+00:00"}