{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:YANSGBPCYTOWDW7PUHRXYISETM","short_pith_number":"pith:YANSGBPC","canonical_record":{"source":{"id":"1111.3419","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-15T02:56:52Z","cross_cats_sorted":[],"title_canon_sha256":"69547ac34d0f4c003a7d92df5c073e942e3e9eb529f810d655d19b312c807ed7","abstract_canon_sha256":"72588886d98d89fa171f29333e9d91bb2b2486b64f9705617418309de80acca0"},"schema_version":"1.0"},"canonical_sha256":"c01b2305e2c4dd61dbefa1e37c22449b3235a777f45803d199ae8c865305be90","source":{"kind":"arxiv","id":"1111.3419","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.3419","created_at":"2026-05-18T03:32:31Z"},{"alias_kind":"arxiv_version","alias_value":"1111.3419v2","created_at":"2026-05-18T03:32:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.3419","created_at":"2026-05-18T03:32:31Z"},{"alias_kind":"pith_short_12","alias_value":"YANSGBPCYTOW","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"YANSGBPCYTOWDW7P","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"YANSGBPC","created_at":"2026-05-18T12:26:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:YANSGBPCYTOWDW7PUHRXYISETM","target":"record","payload":{"canonical_record":{"source":{"id":"1111.3419","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-15T02:56:52Z","cross_cats_sorted":[],"title_canon_sha256":"69547ac34d0f4c003a7d92df5c073e942e3e9eb529f810d655d19b312c807ed7","abstract_canon_sha256":"72588886d98d89fa171f29333e9d91bb2b2486b64f9705617418309de80acca0"},"schema_version":"1.0"},"canonical_sha256":"c01b2305e2c4dd61dbefa1e37c22449b3235a777f45803d199ae8c865305be90","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:32:31.925281Z","signature_b64":"Si/WvpWuOHaZjA2GBC33i8sH9LC7V7LD1xKTtAQOQZn3/bI2YUzuyrgud8ZMSpvd8qEE3YjWBmLuegoZcLjnCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c01b2305e2c4dd61dbefa1e37c22449b3235a777f45803d199ae8c865305be90","last_reissued_at":"2026-05-18T03:32:31.924640Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:32:31.924640Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.3419","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:32:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7B1BA5kltpLBVPAYSBNdYGJLOUso6Yf3/PAGzpq/MRKwg0iCca/zqGZV/0OyguNXRUHXMvLLe9OcHQ9KS8PfCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T22:38:36.258802Z"},"content_sha256":"a0bcc7b2465c648dd74a19a41e7fbc6b270c87c20cd9580f3660484cb57f8b22","schema_version":"1.0","event_id":"sha256:a0bcc7b2465c648dd74a19a41e7fbc6b270c87c20cd9580f3660484cb57f8b22"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:YANSGBPCYTOWDW7PUHRXYISETM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Decomposing Sets of Inversions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lukas Katth\\\"an","submitted_at":"2011-11-15T02:56:52Z","abstract_excerpt":"In this note we consider the question how the set of inversions of a permutation $\\pi \\in S_n$ can be partitioned into two subset, such that those are itself inversion sets of permutations. This is archived by exploiting a connection to a graph theoretical result. For this we establish a correspondence between the substitution decomposition of $\\pi$ and the modular decomposition of its inversion graph. We also consider the special case of multiplicative decompositions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.3419","kind":"arxiv","version":2},"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":"2026-05-18T03:32:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2JBhQfN5bJFCEb2jGooPS6JQd0Y9tI5W3SA5TwIgch1v3N5mjo+GEyWod3plOZ6P7SCegKbW3R6PLkPkl7D2AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T22:38:36.259514Z"},"content_sha256":"e3dc30d24eea37e269b20c62d2df6d40553e5b73183a9a7283ef51d3cec6b37a","schema_version":"1.0","event_id":"sha256:e3dc30d24eea37e269b20c62d2df6d40553e5b73183a9a7283ef51d3cec6b37a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YANSGBPCYTOWDW7PUHRXYISETM/bundle.json","state_url":"https://pith.science/pith/YANSGBPCYTOWDW7PUHRXYISETM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YANSGBPCYTOWDW7PUHRXYISETM/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":"2026-05-31T22:38:36Z","links":{"resolver":"https://pith.science/pith/YANSGBPCYTOWDW7PUHRXYISETM","bundle":"https://pith.science/pith/YANSGBPCYTOWDW7PUHRXYISETM/bundle.json","state":"https://pith.science/pith/YANSGBPCYTOWDW7PUHRXYISETM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YANSGBPCYTOWDW7PUHRXYISETM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:YANSGBPCYTOWDW7PUHRXYISETM","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":"72588886d98d89fa171f29333e9d91bb2b2486b64f9705617418309de80acca0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-15T02:56:52Z","title_canon_sha256":"69547ac34d0f4c003a7d92df5c073e942e3e9eb529f810d655d19b312c807ed7"},"schema_version":"1.0","source":{"id":"1111.3419","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.3419","created_at":"2026-05-18T03:32:31Z"},{"alias_kind":"arxiv_version","alias_value":"1111.3419v2","created_at":"2026-05-18T03:32:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.3419","created_at":"2026-05-18T03:32:31Z"},{"alias_kind":"pith_short_12","alias_value":"YANSGBPCYTOW","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"YANSGBPCYTOWDW7P","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"YANSGBPC","created_at":"2026-05-18T12:26:47Z"}],"graph_snapshots":[{"event_id":"sha256:e3dc30d24eea37e269b20c62d2df6d40553e5b73183a9a7283ef51d3cec6b37a","target":"graph","created_at":"2026-05-18T03:32:31Z","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 note we consider the question how the set of inversions of a permutation $\\pi \\in S_n$ can be partitioned into two subset, such that those are itself inversion sets of permutations. This is archived by exploiting a connection to a graph theoretical result. For this we establish a correspondence between the substitution decomposition of $\\pi$ and the modular decomposition of its inversion graph. We also consider the special case of multiplicative decompositions.","authors_text":"Lukas Katth\\\"an","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-15T02:56:52Z","title":"Decomposing Sets of Inversions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.3419","kind":"arxiv","version":2},"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:a0bcc7b2465c648dd74a19a41e7fbc6b270c87c20cd9580f3660484cb57f8b22","target":"record","created_at":"2026-05-18T03:32:31Z","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":"72588886d98d89fa171f29333e9d91bb2b2486b64f9705617418309de80acca0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-15T02:56:52Z","title_canon_sha256":"69547ac34d0f4c003a7d92df5c073e942e3e9eb529f810d655d19b312c807ed7"},"schema_version":"1.0","source":{"id":"1111.3419","kind":"arxiv","version":2}},"canonical_sha256":"c01b2305e2c4dd61dbefa1e37c22449b3235a777f45803d199ae8c865305be90","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c01b2305e2c4dd61dbefa1e37c22449b3235a777f45803d199ae8c865305be90","first_computed_at":"2026-05-18T03:32:31.924640Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:32:31.924640Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Si/WvpWuOHaZjA2GBC33i8sH9LC7V7LD1xKTtAQOQZn3/bI2YUzuyrgud8ZMSpvd8qEE3YjWBmLuegoZcLjnCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:32:31.925281Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.3419","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a0bcc7b2465c648dd74a19a41e7fbc6b270c87c20cd9580f3660484cb57f8b22","sha256:e3dc30d24eea37e269b20c62d2df6d40553e5b73183a9a7283ef51d3cec6b37a"],"state_sha256":"b2d9822ebbf134f2031614a696269fc59280e7be1deedc9ac37d2cd955aff9f6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IIISY+MkWnzyiVvOqa9DUUEvCa115xfUcvfSdQYmbB5wxu77wDd384VgCqSvinErQgNbusQjVtMeBbk53WPoBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T22:38:36.263379Z","bundle_sha256":"54f85553f3c92a7da0fd694dd140326f52f18d4a80afd4df029f0266931ea11f"}}