{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:KJR4RNZWBVSFODD6FJHX3RHUQ2","short_pith_number":"pith:KJR4RNZW","schema_version":"1.0","canonical_sha256":"5263c8b7360d64570c7e2a4f7dc4f486aa615f91c0dce11d10e1eba11eb68712","source":{"kind":"arxiv","id":"1503.00228","version":1},"attestation_state":"computed","paper":{"title":"Largest minimal inversion-complete and pair-complete sets of permutations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Eric Balandraud (IMJ), Fabio Tardella, Maurice Queyranne","submitted_at":"2015-03-01T07:41:15Z","abstract_excerpt":"We solve two related extremal problems in the theory of permutations. A set $Q$ of permutations of the integers 1 to $n$ is inversion-complete (resp., pair-complete) if for every inversion $(j,i)$, where $1 \\le i \\textless{} j \\le n$, (resp., for every pair $(i,j)$, where $i\\not= j$) there exists a permutation in~$Q$ where $j$ is before~$i$. It is minimally inversion-complete if in addition no proper subset of~$Q$ is inversion-complete; and similarly for pair-completeness. The problems we consider are to determine the maximum cardinality of a minimal inversion-complete set of permutations, and"},"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":"1503.00228","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-03-01T07:41:15Z","cross_cats_sorted":[],"title_canon_sha256":"4fb70109d850b4fec0bae18cfb3d85bcbfb567243ade3ab6ae7ed072554635c6","abstract_canon_sha256":"6f9fc0ae9420820596898f6258034f0b5ebb26e208ae0169629fae18fe9e5df2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:54.856659Z","signature_b64":"jO5Ez0gSmVRw3uTiRM1bOtZAk9v4MOA9gjSnXcvSNpbS+Y1p+ZROHnc1k6PXTiESzjuWhmjg71qQMlgSyIJgAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5263c8b7360d64570c7e2a4f7dc4f486aa615f91c0dce11d10e1eba11eb68712","last_reissued_at":"2026-05-18T02:25:54.856140Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:54.856140Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Largest minimal inversion-complete and pair-complete sets of permutations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Eric Balandraud (IMJ), Fabio Tardella, Maurice Queyranne","submitted_at":"2015-03-01T07:41:15Z","abstract_excerpt":"We solve two related extremal problems in the theory of permutations. A set $Q$ of permutations of the integers 1 to $n$ is inversion-complete (resp., pair-complete) if for every inversion $(j,i)$, where $1 \\le i \\textless{} j \\le n$, (resp., for every pair $(i,j)$, where $i\\not= j$) there exists a permutation in~$Q$ where $j$ is before~$i$. It is minimally inversion-complete if in addition no proper subset of~$Q$ is inversion-complete; and similarly for pair-completeness. The problems we consider are to determine the maximum cardinality of a minimal inversion-complete set of permutations, and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.00228","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1503.00228","created_at":"2026-05-18T02:25:54.856217+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.00228v1","created_at":"2026-05-18T02:25:54.856217+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.00228","created_at":"2026-05-18T02:25:54.856217+00:00"},{"alias_kind":"pith_short_12","alias_value":"KJR4RNZWBVSF","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_16","alias_value":"KJR4RNZWBVSFODD6","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_8","alias_value":"KJR4RNZW","created_at":"2026-05-18T12:29:29.992203+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/KJR4RNZWBVSFODD6FJHX3RHUQ2","json":"https://pith.science/pith/KJR4RNZWBVSFODD6FJHX3RHUQ2.json","graph_json":"https://pith.science/api/pith-number/KJR4RNZWBVSFODD6FJHX3RHUQ2/graph.json","events_json":"https://pith.science/api/pith-number/KJR4RNZWBVSFODD6FJHX3RHUQ2/events.json","paper":"https://pith.science/paper/KJR4RNZW"},"agent_actions":{"view_html":"https://pith.science/pith/KJR4RNZWBVSFODD6FJHX3RHUQ2","download_json":"https://pith.science/pith/KJR4RNZWBVSFODD6FJHX3RHUQ2.json","view_paper":"https://pith.science/paper/KJR4RNZW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.00228&json=true","fetch_graph":"https://pith.science/api/pith-number/KJR4RNZWBVSFODD6FJHX3RHUQ2/graph.json","fetch_events":"https://pith.science/api/pith-number/KJR4RNZWBVSFODD6FJHX3RHUQ2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KJR4RNZWBVSFODD6FJHX3RHUQ2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KJR4RNZWBVSFODD6FJHX3RHUQ2/action/storage_attestation","attest_author":"https://pith.science/pith/KJR4RNZWBVSFODD6FJHX3RHUQ2/action/author_attestation","sign_citation":"https://pith.science/pith/KJR4RNZWBVSFODD6FJHX3RHUQ2/action/citation_signature","submit_replication":"https://pith.science/pith/KJR4RNZWBVSFODD6FJHX3RHUQ2/action/replication_record"}},"created_at":"2026-05-18T02:25:54.856217+00:00","updated_at":"2026-05-18T02:25:54.856217+00:00"}