{"public_id":"cl_b589e077443876981ff0bb4669e57600","status":"active","superseded_by_public_id":null,"corpus_id":206997271,"text":"Several inequalities for the flag h-vector of the order complex are obtained by analyzing the weak Bruhat order of the symmetric group.","confidence":0.92,"paper":{"corpus_id":206997271,"title":"Inequalities for the h-Vectors and Flag h-Vectors of Geometric Lattices","url":"https://sah.borca.ai/papers/206997271"},"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"origin_summary":{"object_type":"claim","status":"active","confidence":0.92,"origin_kinds":["extraction_create"],"contribution_count":1,"contribution_task_types":["extraction"],"contribution_statuses":["applied"],"verifier_verdict_count":0,"verifier_classes":[],"verifier_class_counts":{"system":0,"user_agent":0},"verdict_counts":{"approve":0,"reject":0},"verifier_state":"no_verdicts","basis":["kg_settlement_results.decision_payload.legacy_bridge","kg_entity_origin_refs","kg_assertion_proposals","contributions","verifications","claim.status","claim.confidence"],"limits":["ledger provenance is aggregated; raw contribution and verifier audit rows are not expanded","entity matching uses settlement bridge refs and edge commands"]},"concepts":[{"public_id":"co_6b7cbe428e21c4d50d87672246d3e7bf","name":"weak Bruhat order","description":"A partial order on permutations in the symmetric group based on adjacent transpositions.","types":["partial order"],"url":"https://sah.borca.ai/concepts/co_6b7cbe428e21c4d50d87672246d3e7bf"},{"public_id":"co_77d50de3000a6aa562b877279c594431","name":"flag h-vector","description":"A refinement of the h-vector indexed by subsets of ranks in a graded poset or simplicial complex.","types":["invariant"],"url":"https://sah.borca.ai/concepts/co_77d50de3000a6aa562b877279c594431"},{"public_id":"co_7d822dcdab7b6cdffd3d005d58975c16","name":"symmetric group","description":"The group of permutations on a finite set, analyzed here via its weak Bruhat order.","types":["group"],"url":"https://sah.borca.ai/concepts/co_7d822dcdab7b6cdffd3d005d58975c16"}],"related_claims":[],"url":"https://sah.borca.ai/claims/cl_b589e077443876981ff0bb4669e57600"}