{"id":"https://openalex.org/W2999779318","doi":"https://doi.org/10.37236/8060","title":"On Certain Combinatorial Expansions of the Legendre-Stirling Numbers","display_name":"On Certain Combinatorial Expansions of the Legendre-Stirling Numbers","publication_year":2018,"publication_date":"2018-12-21","ids":{"openalex":"https://openalex.org/W2999779318","doi":"https://doi.org/10.37236/8060","mag":"2999779318"},"language":"en","primary_location":{"id":"doi:10.37236/8060","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8060","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v25i4p57/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/v25i4p57/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5006755219","display_name":"Shi-Mei Ma","orcid":"https://orcid.org/0000-0002-3261-0706"},"institutions":[{"id":"https://openalex.org/I87182695","display_name":"Universidad del Noreste","ror":"https://ror.org/02ahky613","country_code":"MX","type":"education","lineage":["https://openalex.org/I87182695"]}],"countries":["MX"],"is_corresponding":true,"raw_author_name":"Shi-Mei Ma","raw_affiliation_strings":["Northeastern University at Qinhuangdao"],"affiliations":[{"raw_affiliation_string":"Northeastern University at Qinhuangdao","institution_ids":["https://openalex.org/I87182695"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101786936","display_name":"Jun Ma","orcid":"https://orcid.org/0000-0001-7258-9406"},"institutions":[{"id":"https://openalex.org/I183067930","display_name":"Shanghai Jiao Tong University","ror":"https://ror.org/0220qvk04","country_code":"CN","type":"education","lineage":["https://openalex.org/I183067930"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jun Ma","raw_affiliation_strings":["Shanghai jiao tong university"],"affiliations":[{"raw_affiliation_string":"Shanghai jiao tong university","institution_ids":["https://openalex.org/I183067930"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5012604019","display_name":"Yeong\u2010Nan Yeh","orcid":"https://orcid.org/0000-0001-8249-3133"},"institutions":[{"id":"https://openalex.org/I84653119","display_name":"Academia Sinica","ror":"https://ror.org/05bxb3784","country_code":"TW","type":"facility","lineage":["https://openalex.org/I84653119"]}],"countries":["TW"],"is_corresponding":false,"raw_author_name":"Yeong-Nan Yeh","raw_affiliation_strings":["Academia Sinica"],"affiliations":[{"raw_affiliation_string":"Academia Sinica","institution_ids":["https://openalex.org/I84653119"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5006755219"],"corresponding_institution_ids":["https://openalex.org/I87182695"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.35674873,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"25","issue":"4","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":0.9941999912261963,"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/T11428","display_name":"Advanced Mathematical Identities","score":0.9941999912261963,"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":0.9930999875068665,"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/T11191","display_name":"Mathematical functions and polynomials","score":0.9491999745368958,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/legendre-polynomials","display_name":"Legendre polynomials","score":0.8551591634750366},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8060227036476135},{"id":"https://openalex.org/keywords/stirling-numbers-of-the-first-kind","display_name":"Stirling numbers of the first kind","score":0.7667597532272339},{"id":"https://openalex.org/keywords/binomial-coefficient","display_name":"Binomial coefficient","score":0.62208092212677},{"id":"https://openalex.org/keywords/stirling-numbers-of-the-second-kind","display_name":"Stirling numbers of the second kind","score":0.6015263795852661},{"id":"https://openalex.org/keywords/associated-legendre-polynomials","display_name":"Associated Legendre polynomials","score":0.5253143906593323},{"id":"https://openalex.org/keywords/stirling-engine","display_name":"Stirling engine","score":0.49950289726257324},{"id":"https://openalex.org/keywords/legendre-function","display_name":"Legendre function","score":0.49269965291023254},{"id":"https://openalex.org/keywords/generating-function","display_name":"Generating function","score":0.4744490385055542},{"id":"https://openalex.org/keywords/legendre-transformation","display_name":"Legendre transformation","score":0.45835840702056885},{"id":"https://openalex.org/keywords/bell-polynomials","display_name":"Bell polynomials","score":0.45659977197647095},{"id":"https://openalex.org/keywords/combinatorial-proof","display_name":"Combinatorial proof","score":0.4520083963871002},{"id":"https://openalex.org/keywords/stirling-number","display_name":"Stirling number","score":0.4263782799243927},{"id":"https://openalex.org/keywords/continuation","display_name":"Continuation","score":0.42337724566459656},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.410235196352005},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.40691375732421875},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3916494846343994},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1813734471797943},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.14814212918281555},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.07591897249221802},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.0570368766784668},{"id":"https://openalex.org/keywords/gegenbauer-polynomials","display_name":"Gegenbauer polynomials","score":0.049370914697647095}],"concepts":[{"id":"https://openalex.org/C111458787","wikidata":"https://www.wikidata.org/wiki/Q215405","display_name":"Legendre polynomials","level":2,"score":0.8551591634750366},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8060227036476135},{"id":"https://openalex.org/C93765064","wikidata":"https://www.wikidata.org/wiki/Q2657556","display_name":"Stirling numbers of the first kind","level":3,"score":0.7667597532272339},{"id":"https://openalex.org/C34718186","wikidata":"https://www.wikidata.org/wiki/Q209875","display_name":"Binomial coefficient","level":2,"score":0.62208092212677},{"id":"https://openalex.org/C2284495","wikidata":"https://www.wikidata.org/wiki/Q2601117","display_name":"Stirling numbers of the second kind","level":2,"score":0.6015263795852661},{"id":"https://openalex.org/C36956377","wikidata":"https://www.wikidata.org/wiki/Q228376","display_name":"Associated Legendre polynomials","level":5,"score":0.5253143906593323},{"id":"https://openalex.org/C27134321","wikidata":"https://www.wikidata.org/wiki/Q186212","display_name":"Stirling engine","level":2,"score":0.49950289726257324},{"id":"https://openalex.org/C153262748","wikidata":"https://www.wikidata.org/wiki/Q17098449","display_name":"Legendre function","level":3,"score":0.49269965291023254},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.4744490385055542},{"id":"https://openalex.org/C116594300","wikidata":"https://www.wikidata.org/wiki/Q908652","display_name":"Legendre transformation","level":2,"score":0.45835840702056885},{"id":"https://openalex.org/C154485487","wikidata":"https://www.wikidata.org/wiki/Q815666","display_name":"Bell polynomials","level":2,"score":0.45659977197647095},{"id":"https://openalex.org/C118539577","wikidata":"https://www.wikidata.org/wiki/Q3402784","display_name":"Combinatorial proof","level":2,"score":0.4520083963871002},{"id":"https://openalex.org/C168902406","wikidata":"https://www.wikidata.org/wiki/Q80918","display_name":"Stirling number","level":2,"score":0.4263782799243927},{"id":"https://openalex.org/C88626702","wikidata":"https://www.wikidata.org/wiki/Q1128903","display_name":"Continuation","level":2,"score":0.42337724566459656},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.410235196352005},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.40691375732421875},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3916494846343994},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1813734471797943},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.14814212918281555},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.07591897249221802},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0570368766784668},{"id":"https://openalex.org/C78540521","wikidata":"https://www.wikidata.org/wiki/Q1498246","display_name":"Gegenbauer polynomials","level":4,"score":0.049370914697647095},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"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/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/8060","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8060","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v25i4p57/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:10.37236/8060","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8060","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v25i4p57/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":[{"id":"https://openalex.org/G3317480652","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3395846902","display_name":null,"funder_award_id":"11571235","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4418004546","display_name":null,"funder_award_id":"A2017501007","funder_id":"https://openalex.org/F4320322163","funder_display_name":"Natural Science Foundation of Hebei Province"},{"id":"https://openalex.org/G5015758934","display_name":null,"funder_award_id":"11401083","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5994120800","display_name":null,"funder_award_id":"Natural","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8941952471","display_name":null,"funder_award_id":"107-2115-M-001-009-MY3","funder_id":"https://openalex.org/F4320321040","funder_display_name":"National Science Council"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320321040","display_name":"National Science Council","ror":"https://ror.org/02kv4zf79"},{"id":"https://openalex.org/F4320322163","display_name":"Natural Science Foundation of Hebei Province","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2999779318.pdf","grobid_xml":"https://content.openalex.org/works/W2999779318.grobid-xml"},"referenced_works_count":13,"referenced_works":["https://openalex.org/W1662911403","https://openalex.org/W1973856258","https://openalex.org/W2000971050","https://openalex.org/W2019006632","https://openalex.org/W2024768599","https://openalex.org/W2103374220","https://openalex.org/W2123462947","https://openalex.org/W2143170067","https://openalex.org/W2162130471","https://openalex.org/W2949475888","https://openalex.org/W2962915708","https://openalex.org/W6629251564","https://openalex.org/W6815841120"],"related_works":["https://openalex.org/W1515501413","https://openalex.org/W2153676633","https://openalex.org/W2799775110","https://openalex.org/W2081524161","https://openalex.org/W3108596636","https://openalex.org/W2764183459","https://openalex.org/W1644555362","https://openalex.org/W3112509002","https://openalex.org/W3106392023","https://openalex.org/W2804967310"],"abstract_inverted_index":{"The":[0],"Legendre-Stirling":[1,47,58],"numbers":[2,59],"of":[3,17,19,27,30,56,60,65],"the":[4,14,20,28,57],"second":[5],"kind":[6],"were":[7],"introduced":[8],"by":[9],"Everitt":[10],"et":[11],"al.":[12],"in":[13,63],"spectral":[15],"theory":[16],"powers":[18],"Legendre":[21],"differential":[22],"expressions.":[23],"As":[24,50],"a":[25,43],"continuation":[26],"work":[29],"Andrews":[31],"and":[32],"Littlejohn":[33],"(Proc.":[34],"Amer.":[35],"Math.":[36],"Soc.,":[37],"137":[38],"(2009),":[39],"2581-2590),":[40],"we":[41,53],"provide":[42],"combinatorial":[44],"code":[45],"for":[46],"set":[48],"partitions.":[49],"an":[51],"application,":[52],"obtain":[54],"expansions":[55],"both":[61],"kinds":[62],"terms":[64],"binomial":[66],"coefficients.":[67]},"counts_by_year":[{"year":2023,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}