{"id":"https://openalex.org/W2257484642","doi":"https://doi.org/10.1137/16m1059916","title":"A Combinatorial Approach to the Symmetry of $q,t$-Catalan Numbers","display_name":"A Combinatorial Approach to the Symmetry of $q,t$-Catalan Numbers","publication_year":2018,"publication_date":"2018-01-01","ids":{"openalex":"https://openalex.org/W2257484642","doi":"https://doi.org/10.1137/16m1059916","mag":"2257484642"},"language":"en","primary_location":{"id":"doi:10.1137/16m1059916","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1059916","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1602.01126","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5006735153","display_name":"Kyungyong Lee","orcid":"https://orcid.org/0000-0003-0312-4386"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kyungyong Lee","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100361059","display_name":"Li Li","orcid":"https://orcid.org/0000-0001-9102-4232"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Li Li","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5081049009","display_name":"Nicholas A. Loehr","orcid":"https://orcid.org/0000-0003-0684-0310"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Nicholas A. Loehr","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.01609517,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"32","issue":"1","first_page":"191","last_page":"232"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9998999834060669,"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":0.9998999834060669,"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/T11428","display_name":"Advanced Mathematical Identities","score":0.9958999752998352,"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.9890999794006348,"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/catalan-number","display_name":"Catalan number","score":0.8950718641281128},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.851024329662323},{"id":"https://openalex.org/keywords/bijection","display_name":"Bijection","score":0.7748916149139404},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.7195245027542114},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7130496501922607},{"id":"https://openalex.org/keywords/symmetry","display_name":"Symmetry (geometry)","score":0.5904263854026794},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.5640236735343933},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.5306265354156494},{"id":"https://openalex.org/keywords/combinatorial-proof","display_name":"Combinatorial proof","score":0.4465298354625702},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3381471037864685},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.13146531581878662},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.07348436117172241},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.054630786180496216}],"concepts":[{"id":"https://openalex.org/C43929256","wikidata":"https://www.wikidata.org/wiki/Q270513","display_name":"Catalan number","level":2,"score":0.8950718641281128},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.851024329662323},{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.7748916149139404},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.7195245027542114},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7130496501922607},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.5904263854026794},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.5640236735343933},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5306265354156494},{"id":"https://openalex.org/C118539577","wikidata":"https://www.wikidata.org/wiki/Q3402784","display_name":"Combinatorial proof","level":2,"score":0.4465298354625702},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3381471037864685},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.13146531581878662},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.07348436117172241},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.054630786180496216},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/16m1059916","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1059916","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1602.01126","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1602.01126","pdf_url":"https://arxiv.org/pdf/1602.01126","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":"mag:2257484642","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1602.01126.pdf","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":"doi:10.48550/arxiv.1602.01126","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1602.01126","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":"pmh:oai:arXiv.org:1602.01126","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1602.01126","pdf_url":"https://arxiv.org/pdf/1602.01126","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"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G170891774","display_name":null,"funder_award_id":"H98230-16-1-0303","funder_id":"https://openalex.org/F4320311089","funder_display_name":"National Security Agency"},{"id":"https://openalex.org/G4124252003","display_name":null,"funder_award_id":"H98230-14-1-0323","funder_id":"https://openalex.org/F4320311089","funder_display_name":"National Security Agency"}],"funders":[{"id":"https://openalex.org/F4320307070","display_name":"Simmons Family Foundation","ror":"https://ror.org/04278cc17"},{"id":"https://openalex.org/F4320308427","display_name":"American Mathematical Society","ror":"https://ror.org/05vy1kj95"},{"id":"https://openalex.org/F4320310264","display_name":"University of Nebraska-Lincoln","ror":"https://ror.org/043mer456"},{"id":"https://openalex.org/F4320311089","display_name":"National Security Agency","ror":"https://ror.org/0047bvr32"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2257484642.pdf","grobid_xml":"https://content.openalex.org/works/W2257484642.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W1573055504","https://openalex.org/W1596319309","https://openalex.org/W1799589307","https://openalex.org/W2039139573","https://openalex.org/W2048202079","https://openalex.org/W2052753052","https://openalex.org/W2059036144","https://openalex.org/W2060066939","https://openalex.org/W2074692264","https://openalex.org/W2087941871","https://openalex.org/W2101988350","https://openalex.org/W2134984950","https://openalex.org/W2140141642","https://openalex.org/W2319140844","https://openalex.org/W2963339095","https://openalex.org/W3000541016"],"related_works":["https://openalex.org/W2963075274","https://openalex.org/W1597287110","https://openalex.org/W2553922920","https://openalex.org/W1776330888","https://openalex.org/W2073986456","https://openalex.org/W2951218763","https://openalex.org/W1989735024","https://openalex.org/W2997421572","https://openalex.org/W2069221400","https://openalex.org/W3197199649","https://openalex.org/W2963449499","https://openalex.org/W2767360818","https://openalex.org/W1969959148","https://openalex.org/W1598903423","https://openalex.org/W2267159800","https://openalex.org/W1001015058","https://openalex.org/W2888380699","https://openalex.org/W2963466657","https://openalex.org/W1470396398","https://openalex.org/W1599253773"],"abstract_inverted_index":{"The":[0],"$q,t$-Catalan":[1],"numbers":[2,16],"$C_n(q,t)$":[3,39,134],"are":[4,104],"polynomials":[5,20],"in":[6,51,88,133],"$q$":[7],"and":[8,29,33,128],"$t$":[9],"that":[10,79,103,123],"reduce":[11],"to":[12,24,81],"the":[13,62,96,110,131,140],"ordinary":[14],"Catalan":[15],"when":[17],"$q=t=1$.":[18],"These":[19],"have":[21,139],"important":[22],"connections":[23],"representation":[25],"theory,":[26],"algebraic":[27],"geometry,":[28],"symmetric":[30],"functions.":[31],"Haglund":[32],"Haiman":[34],"discovered":[35],"combinatorial":[36,59],"formulas":[37],"for":[38,113,124],"as":[40],"weighted":[41],"sums":[42],"of":[43,61,72,85,98,101,106,135],"Dyck":[44,73],"paths":[45],"(or":[46],"equivalently,":[47],"integer":[48],"partitions":[49,102],"contained":[50],"a":[52,58,82],"staircase":[53],"shape).":[54],"This":[55],"paper":[56],"undertakes":[57],"investigation":[60],"joint":[63,86,111],"symmetry":[64,87,112,142],"property":[65],"$C_n(q,t)=C_n(t,q)$.":[66],"We":[67],"conjecture":[68],"some":[69],"structural":[70],"decompositions":[71],"objects":[74],"into":[75],"\u201cmutually":[76],"opposite\u201d":[77],"subcollections":[78],"lead":[80],"bijective":[83],"explanation":[84],"certain":[89],"cases.":[90],"A":[91],"key":[92],"new":[93],"idea":[94],"is":[95],"construction":[97],"infinite":[99],"chains":[100],"independent":[105],"$n$":[107,115],"but":[108],"induce":[109],"all":[114,129],"simultaneously.":[116],"Using":[117],"these":[118],"methods,":[119],"we":[120],"prove":[121],"combinatorially":[122],"$0\\leq":[125],"k\\leq":[126],"9$":[127],"$n$,":[130],"terms":[132],"total":[136],"degree":[137],"$\\binom{n}{2}-k$":[138],"required":[141],"property.":[143]},"counts_by_year":[],"updated_date":"2026-07-15T18:14:33.161393","created_date":"2025-10-10T00:00:00"}