{"corpus_id":206997271,"paper_sha":"84631b03660a6b90b8eeb0b1f75a7c18ee931424","doi":"10.1007/s00454-004-1137-z","arxiv_id":"math/0405535","pmid":null,"pmcid":null,"mag_id":2074307346,"dblp_id":"journals/dcg/NymanS04","acl_id":null,"title":"Inequalities for the h-Vectors and Flag h-Vectors\nof Geometric Lattices","year":2004,"publication_date":"2004-05-27","venue":"Discrete & Computational Geometry","journal":{"name":"Discrete & Computational Geometry","pages":"533-548","volume":"32"},"journal_issn":null,"journal_title":null,"publication_types":["JournalArticle"],"pubmed_pub_types":null,"s2_fields_of_study":["Mathematics","Computer Science"],"reference_count":21,"citation_count":18,"influential_citation_count":6,"is_open_access":true,"arxiv_categories":["math.CO"],"arxiv_license":null,"arxiv_journal_ref":"Discrete and Computational Geometry, Vol. 32, No. 4, Nov. 2004,\n  pgs 533-548","mesh_headings":null,"chemicals":null,"comments_corrections":null,"source_flags":1,"s2_open_access_pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-004-1137-z.pdf","s2_open_access_landing_url":"https://www.semanticscholar.org/paper/84631b03660a6b90b8eeb0b1f75a7c18ee931424","s2_open_access_license":null,"s2_open_access_status":"BRONZE","pmc_open_access_pdf_url":null,"pmc_open_access_landing_url":null,"pmc_open_access_license":null,"pmc_open_access_status":null,"unpaywall_open_access_pdf_url":null,"unpaywall_open_access_landing_url":null,"unpaywall_open_access_license":null,"unpaywall_open_access_status":null,"abstract":"Abstract\n We prove that the order complex of a geometric lattice has a convex\n ear decomposition. As a consequence, if Δ(L) is the order\n complex of a rank (r + 1) geometric lattice L, then for all i\n ≤ r/2 the h-vector of Δ(L) satisfies hi-1 ≤ hi\n and hi ≤ hr-i.\n \n We also obtain several inequalities for the flag h-vector of\n Δ(L) by analyzing the weak Bruhat order of the symmetric\n group. As an application, we obtain a zonotopal cd-analogue\n of the Dowling–Wilson characterization of geometric lattices which\n minimize Whitney numbers of the second kind. In addition, we are\n able to give a combinatorial flag h-vector proof of hi-1 ≤\n hi when i ≤ (2/7)(r + (5/2)).\n \n","claims":[{"public_id":"cl_1826e70a6012df2b9a996140076105b8","status":"active","text":"A combinatorial flag h-vector proof establishes hi-1 ≤ hi for i ≤ (2/7)(r + (5/2)).","confidence":0.89,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_1826e70a6012df2b9a996140076105b8"},{"public_id":"cl_25cc934c9de46d6c6136e046778d4044","status":"active","text":"A zonotopal cd-analogue of the Dowling–Wilson characterization is obtained for geometric lattices that minimize Whitney numbers of the second kind.","confidence":0.91,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_25cc934c9de46d6c6136e046778d4044"},{"public_id":"cl_cde0e7d3e164e73e1c0da3995b329c46","status":"active","text":"For the order complex of a rank (r + 1) geometric lattice, the h-vector satisfies hi-1 ≤ hi and hi ≤ hr-i for all i ≤ r/2.","confidence":0.98,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_cde0e7d3e164e73e1c0da3995b329c46"},{"public_id":"cl_b589e077443876981ff0bb4669e57600","status":"active","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,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous 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