{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:ONBZPAAAEZXABNF6GC5VU46YIN","short_pith_number":"pith:ONBZPAAA","canonical_record":{"source":{"id":"1009.2895","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-09-15T10:42:29Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"85e3a9b26bfc3009daa97f6b1ad4f49f7b00e17925b1362dd2713318bcf03782","abstract_canon_sha256":"0d3eedee1f817b1ae8d2e04dc59104f8c79ae4d4e7ff51a87bbe48be1c8916d5"},"schema_version":"1.0"},"canonical_sha256":"7343978000266e00b4be30bb5a73d84343679a0f49672d362998e4dcc4a722f1","source":{"kind":"arxiv","id":"1009.2895","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.2895","created_at":"2026-05-18T04:40:55Z"},{"alias_kind":"arxiv_version","alias_value":"1009.2895v1","created_at":"2026-05-18T04:40:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.2895","created_at":"2026-05-18T04:40:55Z"},{"alias_kind":"pith_short_12","alias_value":"ONBZPAAAEZXA","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"ONBZPAAAEZXABNF6","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"ONBZPAAA","created_at":"2026-05-18T12:26:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:ONBZPAAAEZXABNF6GC5VU46YIN","target":"record","payload":{"canonical_record":{"source":{"id":"1009.2895","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-09-15T10:42:29Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"85e3a9b26bfc3009daa97f6b1ad4f49f7b00e17925b1362dd2713318bcf03782","abstract_canon_sha256":"0d3eedee1f817b1ae8d2e04dc59104f8c79ae4d4e7ff51a87bbe48be1c8916d5"},"schema_version":"1.0"},"canonical_sha256":"7343978000266e00b4be30bb5a73d84343679a0f49672d362998e4dcc4a722f1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:55.327066Z","signature_b64":"rf0OR9yGJxdkKnO5NLz7bcd8RGijirsJr3N6bnHoq7pXajU7zBT1O3D6KujNmNi3APInwa5qn4ag/tOq/W2OBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7343978000266e00b4be30bb5a73d84343679a0f49672d362998e4dcc4a722f1","last_reissued_at":"2026-05-18T04:40:55.326042Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:55.326042Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1009.2895","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":"2026-05-18T04:40:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oWnZhNtrNipcQbxXtKw8+x3z5kp3dqUDWuvmQrzshNaD14nHo97IjjjAYnTahg7dV2DQp9vgfn8eAId8HcDzBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T20:16:44.429662Z"},"content_sha256":"dca24714621f57dff4e66ea01bb9cbffce5f713557c61b2abd06d93cbb645297","schema_version":"1.0","event_id":"sha256:dca24714621f57dff4e66ea01bb9cbffce5f713557c61b2abd06d93cbb645297"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:ONBZPAAAEZXABNF6GC5VU46YIN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Betti numbers of smooth Schubert varieties and the remarkable formula of Kostant,Macdonald,Shapiro and Steinberg","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AG","authors_text":"Ersan Akyildiz, James B. Carrell","submitted_at":"2010-09-15T10:42:29Z","abstract_excerpt":"The purpose of this note is to give a refinement of the product formula proved in [1] for the Poincare polynomial of a smooth Schubert variety in the flag variety of an algebraic group G over C. This yields a factorization of the number of elements in a Bruhat interval [e,w] in the Weyl group W of G provided the Schubert variety associated to w is smooth. This gives an elementary necessary condition for a Schubert variety in the flag variety to be smooth."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2895","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":"2026-05-18T04:40:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fCWi5dofC0IOWd/wDq3x4jUV3lO/DN4O8jzqm+o2LdpOMvWi4yQ8QODFdJzkr+7HWCCsTo6p/DWrd1+HQs9AAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T20:16:44.430012Z"},"content_sha256":"0a238d7e71d0df6e23592fcccdb98b7472f4544ecafda45892c53b78a54e42b4","schema_version":"1.0","event_id":"sha256:0a238d7e71d0df6e23592fcccdb98b7472f4544ecafda45892c53b78a54e42b4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ONBZPAAAEZXABNF6GC5VU46YIN/bundle.json","state_url":"https://pith.science/pith/ONBZPAAAEZXABNF6GC5VU46YIN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ONBZPAAAEZXABNF6GC5VU46YIN/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":"2026-07-06T20:16:44Z","links":{"resolver":"https://pith.science/pith/ONBZPAAAEZXABNF6GC5VU46YIN","bundle":"https://pith.science/pith/ONBZPAAAEZXABNF6GC5VU46YIN/bundle.json","state":"https://pith.science/pith/ONBZPAAAEZXABNF6GC5VU46YIN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ONBZPAAAEZXABNF6GC5VU46YIN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:ONBZPAAAEZXABNF6GC5VU46YIN","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":"0d3eedee1f817b1ae8d2e04dc59104f8c79ae4d4e7ff51a87bbe48be1c8916d5","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-09-15T10:42:29Z","title_canon_sha256":"85e3a9b26bfc3009daa97f6b1ad4f49f7b00e17925b1362dd2713318bcf03782"},"schema_version":"1.0","source":{"id":"1009.2895","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.2895","created_at":"2026-05-18T04:40:55Z"},{"alias_kind":"arxiv_version","alias_value":"1009.2895v1","created_at":"2026-05-18T04:40:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.2895","created_at":"2026-05-18T04:40:55Z"},{"alias_kind":"pith_short_12","alias_value":"ONBZPAAAEZXA","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"ONBZPAAAEZXABNF6","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"ONBZPAAA","created_at":"2026-05-18T12:26:12Z"}],"graph_snapshots":[{"event_id":"sha256:0a238d7e71d0df6e23592fcccdb98b7472f4544ecafda45892c53b78a54e42b4","target":"graph","created_at":"2026-05-18T04:40:55Z","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":"The purpose of this note is to give a refinement of the product formula proved in [1] for the Poincare polynomial of a smooth Schubert variety in the flag variety of an algebraic group G over C. This yields a factorization of the number of elements in a Bruhat interval [e,w] in the Weyl group W of G provided the Schubert variety associated to w is smooth. This gives an elementary necessary condition for a Schubert variety in the flag variety to be smooth.","authors_text":"Ersan Akyildiz, James B. Carrell","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-09-15T10:42:29Z","title":"Betti numbers of smooth Schubert varieties and the remarkable formula of Kostant,Macdonald,Shapiro and Steinberg"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2895","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:dca24714621f57dff4e66ea01bb9cbffce5f713557c61b2abd06d93cbb645297","target":"record","created_at":"2026-05-18T04:40:55Z","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":"0d3eedee1f817b1ae8d2e04dc59104f8c79ae4d4e7ff51a87bbe48be1c8916d5","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-09-15T10:42:29Z","title_canon_sha256":"85e3a9b26bfc3009daa97f6b1ad4f49f7b00e17925b1362dd2713318bcf03782"},"schema_version":"1.0","source":{"id":"1009.2895","kind":"arxiv","version":1}},"canonical_sha256":"7343978000266e00b4be30bb5a73d84343679a0f49672d362998e4dcc4a722f1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7343978000266e00b4be30bb5a73d84343679a0f49672d362998e4dcc4a722f1","first_computed_at":"2026-05-18T04:40:55.326042Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:40:55.326042Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rf0OR9yGJxdkKnO5NLz7bcd8RGijirsJr3N6bnHoq7pXajU7zBT1O3D6KujNmNi3APInwa5qn4ag/tOq/W2OBA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:40:55.327066Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.2895","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dca24714621f57dff4e66ea01bb9cbffce5f713557c61b2abd06d93cbb645297","sha256:0a238d7e71d0df6e23592fcccdb98b7472f4544ecafda45892c53b78a54e42b4"],"state_sha256":"2284647845fbf46bfbd4d593d3ede7f7b0c42350e85685106d20197666b89ef6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uUoJy/StS8R9s3eZ9BCT7wLuStUwlRpqeDl3GqtNhIR35yEamPQLL4MUK5sDaWI7ak2TolseTl/AxfRJA4bGCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T20:16:44.431834Z","bundle_sha256":"3da662209a79e179a041f2989faaf6d5c2368fdc35ee39f28d284c372014b34a"}}