{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:TZJ67F45S75UVTRKRUEBEA5ZLN","short_pith_number":"pith:TZJ67F45","canonical_record":{"source":{"id":"1010.0515","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-04T09:26:38Z","cross_cats_sorted":[],"title_canon_sha256":"902f408f9d2be967bbb5f67fc7e4bc335e8c8323b8abdd58ee552062910072c2","abstract_canon_sha256":"280fc4186615a187dc3c9562aad3bc896560991102ec7d376b5471db5221f5a4"},"schema_version":"1.0"},"canonical_sha256":"9e53ef979d97fb4ace2a8d081203b95b7571d6a7ef37ae1981e28536b6dd7ada","source":{"kind":"arxiv","id":"1010.0515","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.0515","created_at":"2026-05-18T04:39:51Z"},{"alias_kind":"arxiv_version","alias_value":"1010.0515v1","created_at":"2026-05-18T04:39:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.0515","created_at":"2026-05-18T04:39:51Z"},{"alias_kind":"pith_short_12","alias_value":"TZJ67F45S75U","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"TZJ67F45S75UVTRK","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"TZJ67F45","created_at":"2026-05-18T12:26:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:TZJ67F45S75UVTRKRUEBEA5ZLN","target":"record","payload":{"canonical_record":{"source":{"id":"1010.0515","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-04T09:26:38Z","cross_cats_sorted":[],"title_canon_sha256":"902f408f9d2be967bbb5f67fc7e4bc335e8c8323b8abdd58ee552062910072c2","abstract_canon_sha256":"280fc4186615a187dc3c9562aad3bc896560991102ec7d376b5471db5221f5a4"},"schema_version":"1.0"},"canonical_sha256":"9e53ef979d97fb4ace2a8d081203b95b7571d6a7ef37ae1981e28536b6dd7ada","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:51.896335Z","signature_b64":"kQcVI+wUHDRLPYl1LPXMNAPkXbR6SObq++VvYj6qNVbz+s6Soqxdzmm+H3hAoGAS+3s9pP5GVRYw4a7YwxRvAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9e53ef979d97fb4ace2a8d081203b95b7571d6a7ef37ae1981e28536b6dd7ada","last_reissued_at":"2026-05-18T04:39:51.895696Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:51.895696Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1010.0515","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":"2026-05-18T04:39:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"D5DBlF2u2ztirw93Vbm33S71qUSu6iCG0FwbHasC0LrrK7PhO1EwFeX4gIhlhO/xlD7AHCJjbOBbrVkuPS7HAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T18:58:38.668299Z"},"content_sha256":"b869cd2b6c3ad5051444dd609d6c4cbb2684513a13650ff7a0b94fa6d814e70d","schema_version":"1.0","event_id":"sha256:b869cd2b6c3ad5051444dd609d6c4cbb2684513a13650ff7a0b94fa6d814e70d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:TZJ67F45S75UVTRKRUEBEA5ZLN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Inversion arrangements and Bruhat intervals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Axel Hultman","submitted_at":"2010-10-04T09:26:38Z","abstract_excerpt":"Let $W$ be a finite reflection group. For a given $w \\in W$, the following assertion may or may not be satisfied:\n  (*) The principal Bruhat order ideal of $w$ contains as many elements as there are regions in the inversion hyperplane arrangement of $w$.\n  We present a type independent combinatorial criterion which characterises the elements $w\\in W$ that satisfy (*). A couple of immediate consequences are derived:\n  (1) The criterion only involves the order ideal of $w$ as an abstract poset. In this sense, (*) is a poset-theoretic property.\n  (2) For $W$ of type $A$, another characterisation "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.0515","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":"2026-05-18T04:39:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yeDQQ1C23WbVP/QnkgaRgGGbFfHcugWMtNQlToO1apBYLl+COzAe/xcRJhIsnCugoR8OgCi6YtCVbBzBTguGCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T18:58:38.668664Z"},"content_sha256":"b20f2320de3f695aaaa9dfa189928e38a678e495f5e37501adfeb1a8ca9add9c","schema_version":"1.0","event_id":"sha256:b20f2320de3f695aaaa9dfa189928e38a678e495f5e37501adfeb1a8ca9add9c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TZJ67F45S75UVTRKRUEBEA5ZLN/bundle.json","state_url":"https://pith.science/pith/TZJ67F45S75UVTRKRUEBEA5ZLN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TZJ67F45S75UVTRKRUEBEA5ZLN/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":"2026-06-22T18:58:38Z","links":{"resolver":"https://pith.science/pith/TZJ67F45S75UVTRKRUEBEA5ZLN","bundle":"https://pith.science/pith/TZJ67F45S75UVTRKRUEBEA5ZLN/bundle.json","state":"https://pith.science/pith/TZJ67F45S75UVTRKRUEBEA5ZLN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TZJ67F45S75UVTRKRUEBEA5ZLN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:TZJ67F45S75UVTRKRUEBEA5ZLN","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":"280fc4186615a187dc3c9562aad3bc896560991102ec7d376b5471db5221f5a4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-04T09:26:38Z","title_canon_sha256":"902f408f9d2be967bbb5f67fc7e4bc335e8c8323b8abdd58ee552062910072c2"},"schema_version":"1.0","source":{"id":"1010.0515","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.0515","created_at":"2026-05-18T04:39:51Z"},{"alias_kind":"arxiv_version","alias_value":"1010.0515v1","created_at":"2026-05-18T04:39:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.0515","created_at":"2026-05-18T04:39:51Z"},{"alias_kind":"pith_short_12","alias_value":"TZJ67F45S75U","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"TZJ67F45S75UVTRK","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"TZJ67F45","created_at":"2026-05-18T12:26:15Z"}],"graph_snapshots":[{"event_id":"sha256:b20f2320de3f695aaaa9dfa189928e38a678e495f5e37501adfeb1a8ca9add9c","target":"graph","created_at":"2026-05-18T04:39:51Z","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":"Let $W$ be a finite reflection group. For a given $w \\in W$, the following assertion may or may not be satisfied:\n  (*) The principal Bruhat order ideal of $w$ contains as many elements as there are regions in the inversion hyperplane arrangement of $w$.\n  We present a type independent combinatorial criterion which characterises the elements $w\\in W$ that satisfy (*). A couple of immediate consequences are derived:\n  (1) The criterion only involves the order ideal of $w$ as an abstract poset. In this sense, (*) is a poset-theoretic property.\n  (2) For $W$ of type $A$, another characterisation ","authors_text":"Axel Hultman","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-04T09:26:38Z","title":"Inversion arrangements and Bruhat intervals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.0515","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:b869cd2b6c3ad5051444dd609d6c4cbb2684513a13650ff7a0b94fa6d814e70d","target":"record","created_at":"2026-05-18T04:39:51Z","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":"280fc4186615a187dc3c9562aad3bc896560991102ec7d376b5471db5221f5a4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-04T09:26:38Z","title_canon_sha256":"902f408f9d2be967bbb5f67fc7e4bc335e8c8323b8abdd58ee552062910072c2"},"schema_version":"1.0","source":{"id":"1010.0515","kind":"arxiv","version":1}},"canonical_sha256":"9e53ef979d97fb4ace2a8d081203b95b7571d6a7ef37ae1981e28536b6dd7ada","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9e53ef979d97fb4ace2a8d081203b95b7571d6a7ef37ae1981e28536b6dd7ada","first_computed_at":"2026-05-18T04:39:51.895696Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:39:51.895696Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kQcVI+wUHDRLPYl1LPXMNAPkXbR6SObq++VvYj6qNVbz+s6Soqxdzmm+H3hAoGAS+3s9pP5GVRYw4a7YwxRvAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:39:51.896335Z","signed_message":"canonical_sha256_bytes"},"source_id":"1010.0515","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b869cd2b6c3ad5051444dd609d6c4cbb2684513a13650ff7a0b94fa6d814e70d","sha256:b20f2320de3f695aaaa9dfa189928e38a678e495f5e37501adfeb1a8ca9add9c"],"state_sha256":"bfc273632dc13f7321bf6a0eceabb7e8b42ad022f4c9199fa3efe88067de021d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LK9dURET3vOE4vFuGJABnH5DNqdYJj+V8dnaKlVnSjwx/6KDu93IPvpPv8T5kofclIXZuhC++80mWXc5OC5PBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T18:58:38.670570Z","bundle_sha256":"b45523d79b514a7439bdcb0962a4e75612b2d9aee26b1ef84c142f6dc4b3dbb1"}}