{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:KWDXRPI4J7EL2BJWIVTA2H5HRI","short_pith_number":"pith:KWDXRPI4","canonical_record":{"source":{"id":"1402.3582","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-14T20:49:01Z","cross_cats_sorted":[],"title_canon_sha256":"3bbee01f6846bcb5c797e2411308af24306efc44ad3cbc65825838cb5ab0ada6","abstract_canon_sha256":"6636645502722761446cc015c2954f11295f57760aaedc9f61f29a44b054548b"},"schema_version":"1.0"},"canonical_sha256":"558778bd1c4fc8bd053645660d1fa78a1337677cdf5c1dfcf96d16fa2c7a6443","source":{"kind":"arxiv","id":"1402.3582","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.3582","created_at":"2026-05-18T02:58:58Z"},{"alias_kind":"arxiv_version","alias_value":"1402.3582v1","created_at":"2026-05-18T02:58:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.3582","created_at":"2026-05-18T02:58:58Z"},{"alias_kind":"pith_short_12","alias_value":"KWDXRPI4J7EL","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_16","alias_value":"KWDXRPI4J7EL2BJW","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_8","alias_value":"KWDXRPI4","created_at":"2026-05-18T12:28:35Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:KWDXRPI4J7EL2BJWIVTA2H5HRI","target":"record","payload":{"canonical_record":{"source":{"id":"1402.3582","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-14T20:49:01Z","cross_cats_sorted":[],"title_canon_sha256":"3bbee01f6846bcb5c797e2411308af24306efc44ad3cbc65825838cb5ab0ada6","abstract_canon_sha256":"6636645502722761446cc015c2954f11295f57760aaedc9f61f29a44b054548b"},"schema_version":"1.0"},"canonical_sha256":"558778bd1c4fc8bd053645660d1fa78a1337677cdf5c1dfcf96d16fa2c7a6443","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:58.392070Z","signature_b64":"9kwNFU5OkYLmVGX66vwynYnmKqExBX9J+YwveKr5JT0d+Mo/czSMZS96/R7C3Weu2HmFi95halx7v52aVYGcDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"558778bd1c4fc8bd053645660d1fa78a1337677cdf5c1dfcf96d16fa2c7a6443","last_reissued_at":"2026-05-18T02:58:58.391146Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:58.391146Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1402.3582","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:58:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SbYLy6uAwTh1j8ZVbuikY08ob0IxNvgmck9Pe3Qzej7W1bbHjCQk0MFhCOU8xTVhyT+YhcC1lXX4ecqeqRfxCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:39:29.793949Z"},"content_sha256":"88febdb61c869e68e2823f26e236b9bfdabe3e291f16261eeee932423ff3f297","schema_version":"1.0","event_id":"sha256:88febdb61c869e68e2823f26e236b9bfdabe3e291f16261eeee932423ff3f297"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:KWDXRPI4J7EL2BJWIVTA2H5HRI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"S-crucial and bicrucial permutations with respect to squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander Konovalov, Ian Gent, Peter Nightingale, Sergey Kitaev, Steve Linton","submitted_at":"2014-02-14T20:49:01Z","abstract_excerpt":"A permutation is square-free if it does not contain two consecutive factors of length two or more that are order-isomorphic. A permutation is bicrucial with respect to squares if it is square-free but any extension of it to the right or to the left by any element gives a permutation that is not square-free.\n  Bicrucial permutations with respect to squares were studied by Avgustinovich et al., who proved that there exist bicrucial permutations of lengths $8k+1, 8k+5, 8k+7$ for $k\\ge 1$. It was left as open questions whether bicrucial permutations of even length, or such permutations of length $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.3582","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:58:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z5n60qa5TFvIGSKiQ03xuVyAo3d/AHqFLeJu+7/ez8pSaiM9Y7YSJ73SnWoRS29ebn5vUdKad1PeZmi6mLwmAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:39:29.794281Z"},"content_sha256":"1993f6500edca4d2bf4834a6d5cf2d1cbd40d8867e143a80d21fbc90bf924a27","schema_version":"1.0","event_id":"sha256:1993f6500edca4d2bf4834a6d5cf2d1cbd40d8867e143a80d21fbc90bf924a27"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KWDXRPI4J7EL2BJWIVTA2H5HRI/bundle.json","state_url":"https://pith.science/pith/KWDXRPI4J7EL2BJWIVTA2H5HRI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KWDXRPI4J7EL2BJWIVTA2H5HRI/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":"2026-06-23T16:39:29Z","links":{"resolver":"https://pith.science/pith/KWDXRPI4J7EL2BJWIVTA2H5HRI","bundle":"https://pith.science/pith/KWDXRPI4J7EL2BJWIVTA2H5HRI/bundle.json","state":"https://pith.science/pith/KWDXRPI4J7EL2BJWIVTA2H5HRI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KWDXRPI4J7EL2BJWIVTA2H5HRI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:KWDXRPI4J7EL2BJWIVTA2H5HRI","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":"6636645502722761446cc015c2954f11295f57760aaedc9f61f29a44b054548b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-14T20:49:01Z","title_canon_sha256":"3bbee01f6846bcb5c797e2411308af24306efc44ad3cbc65825838cb5ab0ada6"},"schema_version":"1.0","source":{"id":"1402.3582","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.3582","created_at":"2026-05-18T02:58:58Z"},{"alias_kind":"arxiv_version","alias_value":"1402.3582v1","created_at":"2026-05-18T02:58:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.3582","created_at":"2026-05-18T02:58:58Z"},{"alias_kind":"pith_short_12","alias_value":"KWDXRPI4J7EL","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_16","alias_value":"KWDXRPI4J7EL2BJW","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_8","alias_value":"KWDXRPI4","created_at":"2026-05-18T12:28:35Z"}],"graph_snapshots":[{"event_id":"sha256:1993f6500edca4d2bf4834a6d5cf2d1cbd40d8867e143a80d21fbc90bf924a27","target":"graph","created_at":"2026-05-18T02:58:58Z","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":"A permutation is square-free if it does not contain two consecutive factors of length two or more that are order-isomorphic. A permutation is bicrucial with respect to squares if it is square-free but any extension of it to the right or to the left by any element gives a permutation that is not square-free.\n  Bicrucial permutations with respect to squares were studied by Avgustinovich et al., who proved that there exist bicrucial permutations of lengths $8k+1, 8k+5, 8k+7$ for $k\\ge 1$. It was left as open questions whether bicrucial permutations of even length, or such permutations of length $","authors_text":"Alexander Konovalov, Ian Gent, Peter Nightingale, Sergey Kitaev, Steve Linton","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-14T20:49:01Z","title":"S-crucial and bicrucial permutations with respect to squares"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.3582","kind":"arxiv","version":1},"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":"sha256:88febdb61c869e68e2823f26e236b9bfdabe3e291f16261eeee932423ff3f297","target":"record","created_at":"2026-05-18T02:58:58Z","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":"6636645502722761446cc015c2954f11295f57760aaedc9f61f29a44b054548b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-14T20:49:01Z","title_canon_sha256":"3bbee01f6846bcb5c797e2411308af24306efc44ad3cbc65825838cb5ab0ada6"},"schema_version":"1.0","source":{"id":"1402.3582","kind":"arxiv","version":1}},"canonical_sha256":"558778bd1c4fc8bd053645660d1fa78a1337677cdf5c1dfcf96d16fa2c7a6443","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"558778bd1c4fc8bd053645660d1fa78a1337677cdf5c1dfcf96d16fa2c7a6443","first_computed_at":"2026-05-18T02:58:58.391146Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:58.391146Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9kwNFU5OkYLmVGX66vwynYnmKqExBX9J+YwveKr5JT0d+Mo/czSMZS96/R7C3Weu2HmFi95halx7v52aVYGcDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:58.392070Z","signed_message":"canonical_sha256_bytes"},"source_id":"1402.3582","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:88febdb61c869e68e2823f26e236b9bfdabe3e291f16261eeee932423ff3f297","sha256:1993f6500edca4d2bf4834a6d5cf2d1cbd40d8867e143a80d21fbc90bf924a27"],"state_sha256":"51834d3987a2c48a2764a7774860a629d2545587914a4415c183968a0f1c18eb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5INFso2Pnwo1m+64VKunDZdlLAlvnxvMwrjqqLrE7CUc1t72ZVEPk4TZ6lcmSZqdzHjq46jf78dD04pIDLRuDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T16:39:29.796203Z","bundle_sha256":"9858552b538ff06f62f350fc6dbfeb3f041c96b87c27be3b6f4697796c2e805f"}}