{"id":"https://openalex.org/W2951015010","doi":"https://doi.org/10.37236/281","title":"A Combinatorial Proof of a Formula for Betti Numbers of a Stacked Polytope","display_name":"A Combinatorial Proof of a Formula for Betti Numbers of a Stacked Polytope","publication_year":2010,"publication_date":"2010-01-05","ids":{"openalex":"https://openalex.org/W2951015010","doi":"https://doi.org/10.37236/281","mag":"2951015010"},"language":"en","primary_location":{"id":"doi:10.37236/281","is_oa":true,"landing_page_url":"https://doi.org/10.37236/281","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v17i1r9/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v17i1r9/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5059295255","display_name":"Suyoung Choi","orcid":"https://orcid.org/0000-0002-3116-1072"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Suyoung Choi","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5036779608","display_name":"Jang Soo Kim","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jang Soo Kim","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5059295255"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.0565,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.77755461,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":96},"biblio":{"volume":"17","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9968000054359436,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/betti-number","display_name":"Betti number","score":0.8405191898345947},{"id":"https://openalex.org/keywords/polytope","display_name":"Polytope","score":0.8101765513420105},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7701312303543091},{"id":"https://openalex.org/keywords/simplicial-complex","display_name":"Simplicial complex","score":0.7655959129333496},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6306877732276917},{"id":"https://openalex.org/keywords/ring","display_name":"Ring (chemistry)","score":0.5499297976493835},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.4945039451122284},{"id":"https://openalex.org/keywords/interpretation","display_name":"Interpretation (philosophy)","score":0.4835050404071808},{"id":"https://openalex.org/keywords/simplicial-approximation-theorem","display_name":"Simplicial approximation theorem","score":0.47032320499420166},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.4619012773036957},{"id":"https://openalex.org/keywords/delta","display_name":"Delta","score":0.45847466588020325},{"id":"https://openalex.org/keywords/combinatorial-proof","display_name":"Combinatorial proof","score":0.44047608971595764},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.21718519926071167},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.16362938284873962},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.14247450232505798},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10747364163398743},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.09404876828193665},{"id":"https://openalex.org/keywords/simplicial-set","display_name":"Simplicial set","score":0.06878489255905151}],"concepts":[{"id":"https://openalex.org/C129621563","wikidata":"https://www.wikidata.org/wiki/Q429593","display_name":"Betti number","level":2,"score":0.8405191898345947},{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.8101765513420105},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7701312303543091},{"id":"https://openalex.org/C187929450","wikidata":"https://www.wikidata.org/wiki/Q994399","display_name":"Simplicial complex","level":2,"score":0.7655959129333496},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6306877732276917},{"id":"https://openalex.org/C2780378348","wikidata":"https://www.wikidata.org/wiki/Q25351438","display_name":"Ring (chemistry)","level":2,"score":0.5499297976493835},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.4945039451122284},{"id":"https://openalex.org/C527412718","wikidata":"https://www.wikidata.org/wiki/Q855395","display_name":"Interpretation (philosophy)","level":2,"score":0.4835050404071808},{"id":"https://openalex.org/C64245159","wikidata":"https://www.wikidata.org/wiki/Q3983972","display_name":"Simplicial approximation theorem","level":5,"score":0.47032320499420166},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.4619012773036957},{"id":"https://openalex.org/C5072461","wikidata":"https://www.wikidata.org/wiki/Q49506","display_name":"Delta","level":2,"score":0.45847466588020325},{"id":"https://openalex.org/C118539577","wikidata":"https://www.wikidata.org/wiki/Q3402784","display_name":"Combinatorial proof","level":2,"score":0.44047608971595764},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.21718519926071167},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.16362938284873962},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.14247450232505798},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10747364163398743},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.09404876828193665},{"id":"https://openalex.org/C155366967","wikidata":"https://www.wikidata.org/wiki/Q1467124","display_name":"Simplicial set","level":4,"score":0.06878489255905151},{"id":"https://openalex.org/C5961521","wikidata":"https://www.wikidata.org/wiki/Q746083","display_name":"Homotopy","level":2,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","level":1,"score":0.0},{"id":"https://openalex.org/C1276947","wikidata":"https://www.wikidata.org/wiki/Q333","display_name":"Astronomy","level":1,"score":0.0},{"id":"https://openalex.org/C79236096","wikidata":"https://www.wikidata.org/wiki/Q14775269","display_name":"Homotopy category","level":3,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.37236/281","is_oa":true,"landing_page_url":"https://doi.org/10.37236/281","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v17i1r9/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:0902.2444","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0902.2444","pdf_url":"https://arxiv.org/pdf/0902.2444","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"mag:2951015010","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/0902.2444","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.247.4161","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.247.4161","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/0902.2444v1.pdf","raw_type":"text"},{"id":"doi:10.48550/arxiv.0902.2444","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.0902.2444","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.37236/281","is_oa":true,"landing_page_url":"https://doi.org/10.37236/281","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v17i1r9/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5706980351","display_name":null,"funder_award_id":"Japanese Society for the Promotion of Science","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"},{"id":"https://openalex.org/G7752643416","display_name":null,"funder_award_id":"Japan","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"}],"funders":[{"id":"https://openalex.org/F4320322348","display_name":"Korea Science and Engineering Foundation","ror":"https://ror.org/013aysd81"},{"id":"https://openalex.org/F4320324161","display_name":"Korea Advanced Institute of Science and Technology","ror":"https://ror.org/05apxxy63"},{"id":"https://openalex.org/F4320334764","display_name":"Japan Society for the Promotion of Science","ror":"https://ror.org/00hhkn466"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2951015010.pdf","grobid_xml":"https://content.openalex.org/works/W2951015010.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W402200086","https://openalex.org/W2751891529"],"related_works":["https://openalex.org/W2999789917","https://openalex.org/W1800855199","https://openalex.org/W2093006702","https://openalex.org/W2950454478","https://openalex.org/W2606445049","https://openalex.org/W2964145535","https://openalex.org/W1967468678","https://openalex.org/W1800038949","https://openalex.org/W2951213213","https://openalex.org/W2951585455","https://openalex.org/W2533812677","https://openalex.org/W2079408601","https://openalex.org/W2962840843","https://openalex.org/W2753167712","https://openalex.org/W2084242186","https://openalex.org/W2276407874","https://openalex.org/W2900113386","https://openalex.org/W2409881747","https://openalex.org/W3098792841","https://openalex.org/W1995549248"],"abstract_inverted_index":{"For":[0],"a":[1,18,23,41],"simplicial":[2],"complex":[3,39],"$\\Delta$,":[4],"the":[5,12,37],"graded":[6],"Betti":[7],"number":[8],"$\\beta_{i,j}({\\bf":[9],"k}[\\Delta])$":[10],"of":[11,40],"Stanley-Reisner":[13],"ring":[14],"${\\bf":[15,20],"k}[\\Delta]$":[16],"over":[17],"field":[19],"k}$":[21],"has":[22],"combinatorial":[24],"interpretation":[25],"due":[26],"to":[27],"Hochster.":[28],"Terai":[29],"and":[30],"Hibi":[31],"showed":[32],"that":[33],"if":[34],"$\\Delta$":[35],"is":[36],"boundary":[38],"$d$-dimensional":[42],"stacked":[43],"polytope":[44],"with":[45],"$n$":[46],"vertices":[47],"for":[48],"$d\\geq3$,":[49],"then":[50],"$\\beta_{k-1,k}({\\bf":[51],"k}[\\Delta])=(k-1){n-d\\choose":[52],"k}$.":[53],"We":[54],"prove":[55],"this":[56],"combinatorially.":[57]},"counts_by_year":[{"year":2013,"cited_by_count":2}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}