{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:HDVEX45RQY5XISWUH3SNRCBEJS","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":"0bdc8798a27e8cf39fe58c35cbd900851dff64bee4abad3a075937e4f9737e21","cross_cats_sorted":["math.MG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-06-27T04:22:49Z","title_canon_sha256":"11be09933a0ac00af5ddc4f460d397c9d15dcf2953b2ee7fe14f85233acecc7b"},"schema_version":"1.0","source":{"id":"2106.14140","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.14140","created_at":"2026-07-05T05:57:29Z"},{"alias_kind":"arxiv_version","alias_value":"2106.14140v2","created_at":"2026-07-05T05:57:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.14140","created_at":"2026-07-05T05:57:29Z"},{"alias_kind":"pith_short_12","alias_value":"HDVEX45RQY5X","created_at":"2026-07-05T05:57:29Z"},{"alias_kind":"pith_short_16","alias_value":"HDVEX45RQY5XISWU","created_at":"2026-07-05T05:57:29Z"},{"alias_kind":"pith_short_8","alias_value":"HDVEX45R","created_at":"2026-07-05T05:57:29Z"}],"graph_snapshots":[{"event_id":"sha256:69718107e556a812def88723ddfbedc7ba0f831e1089972974d25bde7fd721e3","target":"graph","created_at":"2026-07-05T05:57:29Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2106.14140/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Given a set $S$ consisting of $n$ points in $\\mathbb{R}^d$ and one or two vantage points, we study the number of orderings of $S$ induced by measuring the distance (for one vantage point) or the average distance (for two vantage points) from the vantage point(s) to the points of $S$ as the vantage points move through $\\mathbb{R}^d.$ With one vantage point, a theorem of Good and Tideman \\cite{MR505547} shows the maximum number of orderings is a sum of unsigned Stirling numbers of the first kind. We show that the minimum value in all dimensions is $2n-2,$ achieved by $n$ equally spaced points on","authors_text":"Alvaro Carbonero, Beth Anne Castellano, Brittany Ohlinger, Charles Kulick, Gary Gordon, Karie Schmitz","cross_cats":["math.MG"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-06-27T04:22:49Z","title":"Permutations of point sets in $\\mathbb{R}^d$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.14140","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":"sha256:0a227760cd8441c0aab9ab447c11c243bc47e48d078c5fed5b7f91d5737dd672","target":"record","created_at":"2026-07-05T05:57:29Z","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":"0bdc8798a27e8cf39fe58c35cbd900851dff64bee4abad3a075937e4f9737e21","cross_cats_sorted":["math.MG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-06-27T04:22:49Z","title_canon_sha256":"11be09933a0ac00af5ddc4f460d397c9d15dcf2953b2ee7fe14f85233acecc7b"},"schema_version":"1.0","source":{"id":"2106.14140","kind":"arxiv","version":2}},"canonical_sha256":"38ea4bf3b1863b744ad43ee4d888244c8ffad09fff0d7f84befbd0a71270f427","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"38ea4bf3b1863b744ad43ee4d888244c8ffad09fff0d7f84befbd0a71270f427","first_computed_at":"2026-07-05T05:57:29.034403Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:57:29.034403Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"F8j6RWgPavy+Nh52eF9NiC2KOfvLc2uBtwMrj1wUWAKCuamFQI6QVPwnf+DeSWQ7+pn+aut+9uTxZ28gG1sgAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T05:57:29.034820Z","signed_message":"canonical_sha256_bytes"},"source_id":"2106.14140","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0a227760cd8441c0aab9ab447c11c243bc47e48d078c5fed5b7f91d5737dd672","sha256:69718107e556a812def88723ddfbedc7ba0f831e1089972974d25bde7fd721e3"],"state_sha256":"c8aed5f1b2636dcba9b5cb5441fc74191136b4d0f5a29a9ab8bf612915a1cf82"}