{"id":"https://openalex.org/W1599253773","doi":"https://doi.org/10.37236/3432","title":"Rational Associahedra and Noncrossing Partitions","display_name":"Rational Associahedra and Noncrossing Partitions","publication_year":2013,"publication_date":"2013-09-26","ids":{"openalex":"https://openalex.org/W1599253773","doi":"https://doi.org/10.37236/3432","mag":"1599253773"},"language":"en","primary_location":{"id":"doi:10.37236/3432","is_oa":true,"landing_page_url":"https://doi.org/10.37236/3432","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v20i3p54/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/v20i3p54/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5003392168","display_name":"Drew Armstrong","orcid":null},"institutions":[{"id":"https://openalex.org/I145608581","display_name":"University of Miami","ror":"https://ror.org/02dgjyy92","country_code":"US","type":"education","lineage":["https://openalex.org/I145608581"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Drew Armstrong","raw_affiliation_strings":["University of Miami","University of Miami,"],"affiliations":[{"raw_affiliation_string":"University of Miami","institution_ids":["https://openalex.org/I145608581"]},{"raw_affiliation_string":"University of Miami,","institution_ids":["https://openalex.org/I145608581"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5016834944","display_name":"Brendon Rhoades","orcid":"https://orcid.org/0000-0003-0433-5185"},"institutions":[{"id":"https://openalex.org/I36258959","display_name":"University of California, San Diego","ror":"https://ror.org/0168r3w48","country_code":"US","type":"education","lineage":["https://openalex.org/I36258959"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Brendon Rhoades","raw_affiliation_strings":["University of California - San Diego","University of California,  San Diego"],"affiliations":[{"raw_affiliation_string":"University of California - San Diego","institution_ids":["https://openalex.org/I36258959"]},{"raw_affiliation_string":"University of California,  San Diego","institution_ids":["https://openalex.org/I36258959"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5034861890","display_name":"Nathan Williams","orcid":"https://orcid.org/0000-0003-2084-6428"},"institutions":[{"id":"https://openalex.org/I2800403580","display_name":"University of Minnesota System","ror":"https://ror.org/03grvy078","country_code":"US","type":"education","lineage":["https://openalex.org/I2800403580"]},{"id":"https://openalex.org/I130238516","display_name":"University of Minnesota","ror":"https://ror.org/017zqws13","country_code":"US","type":"education","lineage":["https://openalex.org/I130238516"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Nathan Williams","raw_affiliation_strings":["University of Minnesota","University of\u00a0Minnesota"],"affiliations":[{"raw_affiliation_string":"University of Minnesota","institution_ids":["https://openalex.org/I2800403580"]},{"raw_affiliation_string":"University of\u00a0Minnesota","institution_ids":["https://openalex.org/I130238516"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5003392168"],"corresponding_institution_ids":["https://openalex.org/I145608581"],"apc_list":null,"apc_paid":null,"fwci":3.9452,"has_fulltext":true,"cited_by_count":14,"citation_normalized_percentile":{"value":0.92041148,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"20","issue":"3","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":0.9997000098228455,"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":0.9997000098228455,"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":0.9976000189781189,"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/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.996399998664856,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/combinatorics","display_name":"Combinatorics","score":0.8020843863487244},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7248638272285461},{"id":"https://openalex.org/keywords/coprime-integers","display_name":"Coprime integers","score":0.6778550148010254},{"id":"https://openalex.org/keywords/catalan-number","display_name":"Catalan number","score":0.5796486139297485},{"id":"https://openalex.org/keywords/lattice","display_name":"Lattice (music)","score":0.4254749119281769},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.36215701699256897},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.06376445293426514}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8020843863487244},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7248638272285461},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.6778550148010254},{"id":"https://openalex.org/C43929256","wikidata":"https://www.wikidata.org/wiki/Q270513","display_name":"Catalan number","level":2,"score":0.5796486139297485},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.4254749119281769},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36215701699256897},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.06376445293426514},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.37236/3432","is_oa":true,"landing_page_url":"https://doi.org/10.37236/3432","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v20i3p54/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:1305.7286","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1305.7286","pdf_url":"https://arxiv.org/pdf/1305.7286","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":"","raw_type":"text"},{"id":"doi:10.48550/arxiv.1305.7286","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1305.7286","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"},{"id":"mag:1599253773","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"doi:10.37236/3432","is_oa":true,"landing_page_url":"https://doi.org/10.37236/3432","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v20i3p54/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/G4549748060","display_name":"Combinatorics and Representation Theory","funder_award_id":"1068861","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6934431235","display_name":"Combinatorics of Root Systems","funder_award_id":"1001825","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1599253773.pdf","grobid_xml":"https://content.openalex.org/works/W1599253773.grobid-xml"},"referenced_works_count":20,"referenced_works":["https://openalex.org/W179392144","https://openalex.org/W281390074","https://openalex.org/W1634042400","https://openalex.org/W1671066123","https://openalex.org/W1809556225","https://openalex.org/W1811010766","https://openalex.org/W1966420821","https://openalex.org/W1981444921","https://openalex.org/W2049452854","https://openalex.org/W2059036144","https://openalex.org/W2068948808","https://openalex.org/W2132405036","https://openalex.org/W2137493405","https://openalex.org/W2514823897","https://openalex.org/W2616697817","https://openalex.org/W3015657067","https://openalex.org/W3100741585","https://openalex.org/W6602536545","https://openalex.org/W6607309060","https://openalex.org/W6636979980"],"related_works":["https://openalex.org/W2963921456","https://openalex.org/W2798594630","https://openalex.org/W2809530559","https://openalex.org/W2963191685","https://openalex.org/W2952585109","https://openalex.org/W2064164619","https://openalex.org/W2009658434","https://openalex.org/W2951906341","https://openalex.org/W1653293487","https://openalex.org/W1965086851","https://openalex.org/W1771859565","https://openalex.org/W2133341673","https://openalex.org/W2963075274","https://openalex.org/W2733537833","https://openalex.org/W1969959148","https://openalex.org/W2743384536","https://openalex.org/W2267159800","https://openalex.org/W2951218763","https://openalex.org/W2012364034","https://openalex.org/W2485493998"],"abstract_inverted_index":{"Each":[0],"positive":[1,13,31],"rational":[2,32,40,59,95,101,108,127,133],"number":[3,61],"$x>0$":[4,33],"can":[5],"be":[6],"written":[7],"uniquely":[8],"as":[9],"$x=a/(b-a)$":[10],"for":[11,29,91],"coprime":[12],"integers":[14],"$0<a<b$.":[15],"We":[16,80,104,124],"will":[17],"identify":[18],"$x$":[19],"with":[20],"the":[21,39,58,68,119,142,151],"pair":[22],"$(a,b)$.":[23],"In":[24,141],"this":[25],"paper":[26],"we":[27],"define":[28,105,131],"each":[30,158],"a":[34,44,132],"simplicial":[35,46],"complex":[36,47],"$\\mathsf{Ass}(x)=\\mathsf{Ass}(a,b)$":[37],"called":[38],"associahedron.":[41],"It":[42],"is":[43,84],"pure":[45],"of":[48,135,139,154],"dimension":[49],"$a-2$,":[50],"and":[51,65,71,78,86,98],"its":[52,72,92],"maximal":[53],"faces":[54],"are":[55],"counted":[56],"by":[57,76],"Catalan":[60],"$$\\mathsf{Cat}(x)=\\mathsf{Cat}(a,b):=\\frac{(a+b-1)!}{a!\\,b!}.$$The":[62],"cases":[63],"$(a,b)=(n,n+1)$":[64],"$(a,b)=(n,kn+1)$":[66],"recover":[67],"classical":[69],"associahedron":[70],"\"Fuss-Catalan\"":[73],"generalization":[74,134],"studied":[75],"Athanasiadis-Tzanaki":[77],"Fomin-Reading.":[79],"prove":[81],"that":[82],"$\\mathsf{Ass}(a,b)$":[83,106],"shellable":[85],"give":[87],"nice":[88],"product":[89],"formulas":[90],"$h$-vector":[93],"(the":[94,100],"Narayana":[96],"numbers)":[97],"$f$-vector":[99],"Kirkman":[102],"numbers).":[103],"via":[107],"Dyck":[109,128],"paths:":[110],"lattice":[111],"paths":[112,129],"from":[113],"$(0,0)$":[114],"to":[115,130],"$(b,a)$":[116],"staying":[117],"above":[118],"line":[120],"$y":[121],"=":[122,145],"\\frac{a}{b}x$.":[123],"also":[125],"use":[126],"noncrossing":[136,152],"perfect":[137],"matchings":[138],"$[2n]$.":[140],"case":[143],"$(a,b)":[144],"(n,":[146],"mn+1)$,":[147],"our":[148],"construction":[149],"produces":[150],"partitions":[153],"$[(m+1)n]$":[155],"in":[156],"which":[157],"block":[159],"has":[160],"size":[161],"$m+1$.":[162]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":4}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}