{"id":"https://openalex.org/W7204586991","doi":"https://doi.org/10.48550/arxiv.2608.27126","title":"Counting Boolean Antichains","display_name":"Counting Boolean Antichains","publication_year":2026,"publication_date":"2026-08-27","ids":{"openalex":"https://openalex.org/W7204586991","doi":"https://doi.org/10.48550/arxiv.2608.27126"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.27126","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.27126","pdf_url":"https://arxiv.org/pdf/2608.27126","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.27126","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5148489567","display_name":"Julien Garber","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Garber, Julien","raw_affiliation_strings":["D.H.)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"D.H.)","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5148554517","display_name":"Lina S. Goltermann","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I904495901","display_name":"Ruhr University Bochum","ror":"https://ror.org/04tsk2644","country_code":"DE","type":"education","lineage":["https://openalex.org/I904495901"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Goltermann, Lina S.","raw_affiliation_strings":["Fakult\u00e4t f\u00fcr Mathematik , Ruhr-Universit\u00e4t Bochum , Universit\u00e4tstr. 150 , 44801 Bochum ("],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Fakult\u00e4t f\u00fcr Mathematik , Ruhr-Universit\u00e4t Bochum , Universit\u00e4tstr. 150 , 44801 Bochum (","institution_ids":["https://openalex.org/I904495901"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5140369463","display_name":"Daniel Horiatakis","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I904495901","display_name":"Ruhr University Bochum","ror":"https://ror.org/04tsk2644","country_code":"DE","type":"education","lineage":["https://openalex.org/I904495901"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Horiatakis, Daniel","raw_affiliation_strings":["Fakult\u00e4t f\u00fcr Mathematik , Ruhr-Universit\u00e4t Bochum , Universit\u00e4tstr. 150 , 44801 Bochum ("],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Fakult\u00e4t f\u00fcr Mathematik , Ruhr-Universit\u00e4t Bochum , Universit\u00e4tstr. 150 , 44801 Bochum (","institution_ids":["https://openalex.org/I904495901"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5111768089","display_name":"Dennis K\u00f6nig","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I904495901","display_name":"Ruhr University Bochum","ror":"https://ror.org/04tsk2644","country_code":"DE","type":"education","lineage":["https://openalex.org/I904495901"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"K\u00f6nig, Dennis","raw_affiliation_strings":["Fakult\u00e4t f\u00fcr Mathematik , Ruhr-Universit\u00e4t Bochum , Universit\u00e4tstr. 150 , 44801 Bochum ("],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Fakult\u00e4t f\u00fcr Mathematik , Ruhr-Universit\u00e4t Bochum , Universit\u00e4tstr. 150 , 44801 Bochum (","institution_ids":["https://openalex.org/I904495901"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5099137359","display_name":"Tal Gottesman","orcid":"https://orcid.org/0009-0009-8314-1178","observed_orcids":["https://orcid.org/0009-0009-8314-1178"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Gottesman, Tal","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":1.0,"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"}},"topics":[{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":1.0,"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/T11727","display_name":"Advanced Algebra and Logic","score":0.0,"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"}},{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":0.0,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/boolean-lattice","display_name":"Boolean lattice","score":0.8640000224113464},{"id":"https://openalex.org/keywords/partition-lattice","display_name":"partition lattice","score":0.9869999885559082},{"id":"https://openalex.org/keywords/tamari-lattice","display_name":"Tamari lattice","score":0.9909999966621399},{"id":"https://openalex.org/keywords/incidence-algebras","display_name":"incidence algebras","score":0.7710000276565552},{"id":"https://openalex.org/keywords/lattice-theory","display_name":"lattice theory","score":0.47600001096725464}],"concepts":[{"id":"https://openalex.org/C136134351","wikidata":"https://www.wikidata.org/wiki/Q431937","display_name":"Antichain","level":3,"score":0.8648999929428101},{"id":"https://openalex.org/C207764348","wikidata":"https://www.wikidata.org/wiki/Q7858706","display_name":"Two-element Boolean algebra","level":4,"score":0.7146000266075134},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6216999888420105},{"id":"https://openalex.org/C81518389","wikidata":"https://www.wikidata.org/wiki/Q5156447","display_name":"Complete Boolean algebra","level":5,"score":0.5896999835968018},{"id":"https://openalex.org/C45503250","wikidata":"https://www.wikidata.org/wiki/Q4943350","display_name":"Boolean algebras canonically defined","level":5,"score":0.5789999961853027},{"id":"https://openalex.org/C158465420","wikidata":"https://www.wikidata.org/wiki/Q1979515","display_name":"Boolean expression","level":3,"score":0.5340999960899353},{"id":"https://openalex.org/C10021369","wikidata":"https://www.wikidata.org/wiki/Q1166774","display_name":"Stone's representation theorem for Boolean algebras","level":5,"score":0.5231000185012817},{"id":"https://openalex.org/C68481662","wikidata":"https://www.wikidata.org/wiki/Q176197","display_name":"Standard Boolean model","level":5,"score":0.517799973487854},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5105999708175659},{"id":"https://openalex.org/C188183281","wikidata":"https://www.wikidata.org/wiki/Q2898309","display_name":"Parity function","level":4,"score":0.49070000648498535},{"id":"https://openalex.org/C39685927","wikidata":"https://www.wikidata.org/wiki/Q173183","display_name":"Boolean algebra","level":2,"score":0.4771000146865845},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.47530001401901245},{"id":"https://openalex.org/C188159880","wikidata":"https://www.wikidata.org/wiki/Q5499761","display_name":"Free Boolean algebra","level":5,"score":0.44369998574256897},{"id":"https://openalex.org/C111797529","wikidata":"https://www.wikidata.org/wiki/Q17080584","display_name":"Maximum satisfiability problem","level":3,"score":0.43299999833106995},{"id":"https://openalex.org/C9429942","wikidata":"https://www.wikidata.org/wiki/Q1781698","display_name":"Product term","level":5,"score":0.4318000078201294},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.41990000009536743},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.41589999198913574},{"id":"https://openalex.org/C4926405","wikidata":"https://www.wikidata.org/wiki/Q3269980","display_name":"Boolean domain","level":5,"score":0.4115999937057495},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.38670000433921814},{"id":"https://openalex.org/C196836630","wikidata":"https://www.wikidata.org/wiki/Q4753279","display_name":"And-inverter graph","level":4,"score":0.35679998993873596},{"id":"https://openalex.org/C180645754","wikidata":"https://www.wikidata.org/wiki/Q474715","display_name":"Partially ordered set","level":2,"score":0.33309999108314514},{"id":"https://openalex.org/C2780801425","wikidata":"https://www.wikidata.org/wiki/Q5164392","display_name":"Construct (python library)","level":2,"score":0.3273000121116638},{"id":"https://openalex.org/C141796577","wikidata":"https://www.wikidata.org/wiki/Q837479","display_name":"Boolean circuit","level":3,"score":0.3208000063896179},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.303600013256073},{"id":"https://openalex.org/C200415742","wikidata":"https://www.wikidata.org/wiki/Q1350285","display_name":"True quantified Boolean formula","level":2,"score":0.2800999879837036},{"id":"https://openalex.org/C134444547","wikidata":"https://www.wikidata.org/wiki/Q585230","display_name":"Boolean network","level":3,"score":0.2759999930858612},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.26930001378059387},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.2621000111103058},{"id":"https://openalex.org/C2776639384","wikidata":"https://www.wikidata.org/wiki/Q840396","display_name":"Ideal (ethics)","level":2,"score":0.25839999318122864}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.27126","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.27126","pdf_url":"https://arxiv.org/pdf/2608.27126","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.27126","doi":"https://doi.org/10.48550/arxiv.2608.27126","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.27126","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.27126","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.27126","pdf_url":"https://arxiv.org/pdf/2608.27126","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204586991.pdf","grobid_xml":"https://content.openalex.org/works/W7204586991.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W4416019919","https://openalex.org/W7140325892"],"related_works":["https://openalex.org/W7204691160","https://openalex.org/W7204842554","https://openalex.org/W4377003528","https://openalex.org/W4408444226","https://openalex.org/W2003724356","https://openalex.org/W2474757319","https://openalex.org/W4385832539","https://openalex.org/W3170930009","https://openalex.org/W4400088323","https://openalex.org/W7104370622"],"abstract_inverted_index":{"We":[0],"say":[1],"that":[2],"an":[3],"antichain":[4],"in":[5,17,19,31,76,89],"a":[6,14,20,29,84],"lattice":[7,41],"$L$":[8,18],"is":[9,65],"Boolean":[10,15,74,87,90,111],"if":[11],"it":[12,64],"generates":[13],"sublattice":[16],"particularly":[21],"nice":[22],"way.":[23],"These":[24],"purely":[25],"combinatorial":[26],"objects":[27],"play":[28],"role":[30],"the":[32,36,40,53,100,107],"representation":[33],"theory":[34],"of":[35,39,48,52,86,109],"incidence":[37],"algebra":[38],"$L$,":[42],"as":[43],"indicated":[44],"by":[45],"recent":[46],"results":[47],"Rognerud,":[49],"Y\u0131ld\u0131r\u0131m":[50],"and":[51,59,72,94],"last":[54],"author":[55],"with":[56],"Kl\u00e1sz,":[57],"Kleinau":[58],"Marczinzik.":[60],"Given":[61],"these":[62,110],"motivations,":[63],"natural":[66],"to":[67],"ask:":[68],"Can":[69],"we":[70,82,98,104],"count":[71],"construct":[73],"antichains":[75,88],"certain":[77],"lattices?":[78],"In":[79],"this":[80],"paper":[81],"give":[83],"construction":[85],"lattices,":[91],"partition":[92],"lattices":[93],"Tamari":[95],"lattices.":[96],"Furthermore,":[97],"share":[99],"surprisingly":[101],"elegant":[102],"formulas":[103],"found":[105],"for":[106],"number":[108],"antichains.":[112]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-08-29T00:00:00"}