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Relationships with point-line incidences in the plane, Bruhat order of permutations, and $TP$ completability are also presented. We also examine the number and positionings of equal $2\\textrm{-by-}2$ minors in a $2\\textrm{-by-}n$ $TP$ matrix, and give a relationship between the location of equal $2\\textrm{-by-}2$ minors and outerplanar graphs."},"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":"1309.4186","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-17T05:27:48Z","cross_cats_sorted":[],"title_canon_sha256":"4e0ef703d0074d895654276a682e87196fccfa0aae5c67d63c31e0ddce4addf5","abstract_canon_sha256":"9ef8b965252a7056052ace8dc86b92c2f719509c40ee45dfa066bca8ef9d8670"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:09.350646Z","signature_b64":"U0gn/44ExMWPnBPuXkh484ZED3dlE+RaSHFcr9ui3mT800ck4CYQrw8cjunNGiUfa8wEtXigxe9BgsuBo101Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"45d8d7db7f63f89fb6f35dc8963469eaea0dea92b08d59cde7afef2c0396fbfc","last_reissued_at":"2026-05-18T03:13:09.349947Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:09.349947Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Equal Entries in Totally Positive Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Charles R. 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We also examine the number and positionings of equal $2\\textrm{-by-}2$ minors in a $2\\textrm{-by-}n$ $TP$ matrix, and give a relationship between the location of equal $2\\textrm{-by-}2$ minors and outerplanar graphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.4186","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":"1309.4186","created_at":"2026-05-18T03:13:09.350056+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.4186v1","created_at":"2026-05-18T03:13:09.350056+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.4186","created_at":"2026-05-18T03:13:09.350056+00:00"},{"alias_kind":"pith_short_12","alias_value":"IXMNPW37MP4J","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_16","alias_value":"IXMNPW37MP4J7NXT","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_8","alias_value":"IXMNPW37","created_at":"2026-05-18T12:27:49.015174+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/IXMNPW37MP4J7NXTLXEJMNDJ5L","json":"https://pith.science/pith/IXMNPW37MP4J7NXTLXEJMNDJ5L.json","graph_json":"https://pith.science/api/pith-number/IXMNPW37MP4J7NXTLXEJMNDJ5L/graph.json","events_json":"https://pith.science/api/pith-number/IXMNPW37MP4J7NXTLXEJMNDJ5L/events.json","paper":"https://pith.science/paper/IXMNPW37"},"agent_actions":{"view_html":"https://pith.science/pith/IXMNPW37MP4J7NXTLXEJMNDJ5L","download_json":"https://pith.science/pith/IXMNPW37MP4J7NXTLXEJMNDJ5L.json","view_paper":"https://pith.science/paper/IXMNPW37","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.4186&json=true","fetch_graph":"https://pith.science/api/pith-number/IXMNPW37MP4J7NXTLXEJMNDJ5L/graph.json","fetch_events":"https://pith.science/api/pith-number/IXMNPW37MP4J7NXTLXEJMNDJ5L/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IXMNPW37MP4J7NXTLXEJMNDJ5L/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IXMNPW37MP4J7NXTLXEJMNDJ5L/action/storage_attestation","attest_author":"https://pith.science/pith/IXMNPW37MP4J7NXTLXEJMNDJ5L/action/author_attestation","sign_citation":"https://pith.science/pith/IXMNPW37MP4J7NXTLXEJMNDJ5L/action/citation_signature","submit_replication":"https://pith.science/pith/IXMNPW37MP4J7NXTLXEJMNDJ5L/action/replication_record"}},"created_at":"2026-05-18T03:13:09.350056+00:00","updated_at":"2026-05-18T03:13:09.350056+00:00"}