{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:HXKJXDQSW57I5O4WWXRKOG5DXZ","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":"8dfed303b5679bc51d2d0f12a3e6d5cbad6b5040246c88044b24a916b8736575","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-07T17:51:24Z","title_canon_sha256":"e79fde57e10fa1c09f9eac1191d2a225f9e8256ec015056c4b781feaf92272bc"},"schema_version":"1.0","source":{"id":"1006.1312","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1006.1312","created_at":"2026-05-18T04:28:45Z"},{"alias_kind":"arxiv_version","alias_value":"1006.1312v2","created_at":"2026-05-18T04:28:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.1312","created_at":"2026-05-18T04:28:45Z"},{"alias_kind":"pith_short_12","alias_value":"HXKJXDQSW57I","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"HXKJXDQSW57I5O4W","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"HXKJXDQS","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:d2df58b19556f778437a21c5185594cbe3bd588aab10c57f2e38cfbaf0efab1c","target":"graph","created_at":"2026-05-18T04:28:45Z","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":"This paper introduces two matrix analogues for set partitions. A composition matrix on a finite set X is an upper triangular matrix whose entries partition X, and for which there are no rows or columns containing only empty sets. A partition matrix is a composition matrix in which an order is placed on where entries may appear relative to one-another. We show that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classi","authors_text":"Anders Claesson, Mark Dukes, Martina Kubitzke","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-07T17:51:24Z","title":"Partition and composition matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.1312","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:a451b91e0532925e39d5d2b5c25db40b23fb9b19eaf3f762247d979ede38a035","target":"record","created_at":"2026-05-18T04:28:45Z","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":"8dfed303b5679bc51d2d0f12a3e6d5cbad6b5040246c88044b24a916b8736575","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-07T17:51:24Z","title_canon_sha256":"e79fde57e10fa1c09f9eac1191d2a225f9e8256ec015056c4b781feaf92272bc"},"schema_version":"1.0","source":{"id":"1006.1312","kind":"arxiv","version":2}},"canonical_sha256":"3dd49b8e12b77e8ebb96b5e2a71ba3be4f597f8ae5148995740268c645e1532c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3dd49b8e12b77e8ebb96b5e2a71ba3be4f597f8ae5148995740268c645e1532c","first_computed_at":"2026-05-18T04:28:45.628072Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:28:45.628072Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0dP7QB64VxtVnKwxVEqYdZdAF/Xni+Do4llWBCk+cCgLzgJ+u7LtIq1sAB3VOa3VAnAFuRoEQlRiUbzXNkk3Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:28:45.628691Z","signed_message":"canonical_sha256_bytes"},"source_id":"1006.1312","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a451b91e0532925e39d5d2b5c25db40b23fb9b19eaf3f762247d979ede38a035","sha256:d2df58b19556f778437a21c5185594cbe3bd588aab10c57f2e38cfbaf0efab1c"],"state_sha256":"c000a303001373ecf49d88504cd28bc4c527ad3db44f4cd38a3ef228ebadbaf2"}