{"id":"https://openalex.org/W2999197910","doi":"https://doi.org/10.37236/8870","title":"On Parity Unimodality of $q$-Catalan Polynomials","display_name":"On Parity Unimodality of $q$-Catalan Polynomials","publication_year":2020,"publication_date":"2020-01-10","ids":{"openalex":"https://openalex.org/W2999197910","doi":"https://doi.org/10.37236/8870","mag":"2999197910"},"language":"en","primary_location":{"id":"doi:10.37236/8870","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8870","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i1p3/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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i1p3/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5049446116","display_name":"Guoce Xin","orcid":"https://orcid.org/0000-0001-6259-1385"},"institutions":[{"id":"https://openalex.org/I96852419","display_name":"Capital Normal University","ror":"https://ror.org/005edt527","country_code":"CN","type":"education","lineage":["https://openalex.org/I96852419"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Guoce Xin","raw_affiliation_strings":["Capital Normal University"],"affiliations":[{"raw_affiliation_string":"Capital Normal University","institution_ids":["https://openalex.org/I96852419"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5044856007","display_name":"Yueming Zhong","orcid":"https://orcid.org/0000-0003-2867-6654"},"institutions":[{"id":"https://openalex.org/I96852419","display_name":"Capital Normal University","ror":"https://ror.org/005edt527","country_code":"CN","type":"education","lineage":["https://openalex.org/I96852419"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yueming Zhong","raw_affiliation_strings":["Capital Normal University"],"affiliations":[{"raw_affiliation_string":"Capital Normal University","institution_ids":["https://openalex.org/I96852419"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5049446116"],"corresponding_institution_ids":["https://openalex.org/I96852419"],"apc_list":null,"apc_paid":null,"fwci":1.9259,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.84403472,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"27","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9904999732971191,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9904999732971191,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/unimodality","display_name":"Unimodality","score":0.9673279523849487},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8834987878799438},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8084938526153564},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.7593024373054504},{"id":"https://openalex.org/keywords/coprime-integers","display_name":"Coprime integers","score":0.669773519039154},{"id":"https://openalex.org/keywords/parity","display_name":"Parity (physics)","score":0.646016538143158},{"id":"https://openalex.org/keywords/catalan-number","display_name":"Catalan number","score":0.5791558623313904},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.38715100288391113},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.047247081995010376}],"concepts":[{"id":"https://openalex.org/C21596040","wikidata":"https://www.wikidata.org/wiki/Q2896771","display_name":"Unimodality","level":2,"score":0.9673279523849487},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8834987878799438},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8084938526153564},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.7593024373054504},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.669773519039154},{"id":"https://openalex.org/C2777151079","wikidata":"https://www.wikidata.org/wiki/Q141160","display_name":"Parity (physics)","level":2,"score":0.646016538143158},{"id":"https://openalex.org/C43929256","wikidata":"https://www.wikidata.org/wiki/Q270513","display_name":"Catalan number","level":2,"score":0.5791558623313904},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.38715100288391113},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.047247081995010376},{"id":"https://openalex.org/C109214941","wikidata":"https://www.wikidata.org/wiki/Q18334","display_name":"Particle physics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/8870","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8870","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i1p3/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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.37236/8870","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8870","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i1p3/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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2999197910.pdf","grobid_xml":"https://content.openalex.org/works/W2999197910.grobid-xml"},"referenced_works_count":32,"referenced_works":["https://openalex.org/W1509692083","https://openalex.org/W1547955525","https://openalex.org/W1551177513","https://openalex.org/W1969719825","https://openalex.org/W1978565415","https://openalex.org/W1981750187","https://openalex.org/W1982210080","https://openalex.org/W1990623507","https://openalex.org/W1994500535","https://openalex.org/W2008006321","https://openalex.org/W2022703028","https://openalex.org/W2044185619","https://openalex.org/W2057506387","https://openalex.org/W2082899356","https://openalex.org/W2097357107","https://openalex.org/W2271841861","https://openalex.org/W2300684981","https://openalex.org/W2344834703","https://openalex.org/W2596435438","https://openalex.org/W2734752527","https://openalex.org/W2808490087","https://openalex.org/W2905227611","https://openalex.org/W2962710165","https://openalex.org/W2962953261","https://openalex.org/W3000541016","https://openalex.org/W4238933498","https://openalex.org/W4239240946","https://openalex.org/W6632921768","https://openalex.org/W6652189947","https://openalex.org/W6654375628","https://openalex.org/W6666120698","https://openalex.org/W6734956340"],"related_works":["https://openalex.org/W2373855227","https://openalex.org/W2075968040","https://openalex.org/W2010176972","https://openalex.org/W27154759","https://openalex.org/W2054787334","https://openalex.org/W2076454667","https://openalex.org/W94338811","https://openalex.org/W2951233655","https://openalex.org/W3025068760","https://openalex.org/W3180944311"],"abstract_inverted_index":{"A":[0],"polynomial":[1],"$A(q)=\\sum_{i=0}^n":[2],"a_iq^i$":[3],"is":[4,29,60],"said":[5],"to":[6,31,55],"be":[7,32],"unimodal":[8,52],"if":[9],"$a_0\\leqslant":[10],"a_1\\leqslant":[11],"\\cdots":[12,17],"\\leqslant":[13],"a_k\\geqslant":[14],"a_{k+1}":[15],"\\geqslant":[16,18],"a_n$.":[19],"We":[20,47],"investigate":[21],"the":[22,67],"unimodality":[23],"of":[24,43],"rational":[25],"$q$-Catalan":[26],"polynomials,":[27],"which":[28],"defined":[30],"$C_{m,n}(q)=":[33],"\\frac{1}{[n+m]}":[34],"{m+n":[35],"\\brack":[36],"n}_q":[37],"$":[38],"for":[39,76],"a":[40,80],"coprime":[41],"pair":[42],"positive":[44],"integers":[45],"$(m,n)$.":[46],"conjecture":[48,75],"that":[49],"they":[50],"are":[51],"with":[53],"respect":[54],"parity,":[56],"or":[57],"equivalently,":[58],"$(1+q)C_{m+n}(q)$":[59],"unimodal.":[61],"By":[62],"using":[63],"generating":[64],"functions":[65],"and":[66],"constant":[68],"term":[69],"method,":[70],"we":[71],"verify":[72],"our":[73,74],"$m\\le":[77],"5$":[78],"in":[79],"straightforward":[81],"way.":[82]},"counts_by_year":[{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}