{"id":"https://openalex.org/W2981369365","doi":"https://doi.org/10.1142/s0218196720500496","title":"On the enumeration and asymptotic growth of free quasigroup words","display_name":"On the enumeration and asymptotic growth of free quasigroup words","publication_year":2020,"publication_date":"2020-06-18","ids":{"openalex":"https://openalex.org/W2981369365","doi":"https://doi.org/10.1142/s0218196720500496","mag":"2981369365"},"language":"en","primary_location":{"id":"doi:10.1142/s0218196720500496","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218196720500496","pdf_url":null,"source":{"id":"https://openalex.org/S23260266","display_name":"International Journal of Algebra and Computation","issn_l":"0218-1967","issn":["0218-1967","1793-6500"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Algebra and Computation","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1910.09749","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5068463429","display_name":"Jonathan D. H. Smith","orcid":"https://orcid.org/0000-0002-2075-0744"},"institutions":[{"id":"https://openalex.org/I173911158","display_name":"Iowa State University","ror":"https://ror.org/04rswrd78","country_code":"US","type":"education","lineage":["https://openalex.org/I173911158"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Jonathan D. H. Smith","raw_affiliation_strings":["Department of Mathematics, Iowa State University, Ames, IA 50011, USA","Department of Mathematics, Iowa State University, Ames, IA-50011, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Iowa State University, Ames, IA 50011, USA","institution_ids":["https://openalex.org/I173911158"]},{"raw_affiliation_string":"Department of Mathematics, Iowa State University, Ames, IA-50011, USA","institution_ids":["https://openalex.org/I173911158"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5074396116","display_name":"Stefanie G. Wang","orcid":null},"institutions":[{"id":"https://openalex.org/I202524275","display_name":"Smith College","ror":"https://ror.org/0497crr92","country_code":"US","type":"education","lineage":["https://openalex.org/I202524275"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Stefanie G. Wang","raw_affiliation_strings":["Department of Mathematics & Statistics, Smith College, Northampton, MA 01063, USA","Department of Mathematics and Statistics, Smith College, Northampton, MA, 01063, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics & Statistics, Smith College, Northampton, MA 01063, USA","institution_ids":["https://openalex.org/I202524275"]},{"raw_affiliation_string":"Department of Mathematics and Statistics, Smith College, Northampton, MA, 01063, USA","institution_ids":["https://openalex.org/I202524275"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5068463429"],"corresponding_institution_ids":["https://openalex.org/I173911158"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.1063283,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"30","issue":"07","first_page":"1465","last_page":"1483"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9994000196456909,"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":0.9994000196456909,"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/T11567","display_name":"semigroups and automata theory","score":0.9980999827384949,"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/T11797","display_name":"graph theory and CDMA systems","score":0.9957000017166138,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/quasigroup","display_name":"Quasigroup","score":0.946333646774292},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7337755560874939},{"id":"https://openalex.org/keywords/catalan-number","display_name":"Catalan number","score":0.6419904828071594},{"id":"https://openalex.org/keywords/enumeration","display_name":"Enumeration","score":0.5542621612548828},{"id":"https://openalex.org/keywords/catalan","display_name":"Catalan","score":0.5397335886955261},{"id":"https://openalex.org/keywords/magma","display_name":"Magma","score":0.5208585858345032},{"id":"https://openalex.org/keywords/euclidean-algorithm","display_name":"Euclidean algorithm","score":0.5061778426170349},{"id":"https://openalex.org/keywords/word","display_name":"Word (group theory)","score":0.48444512486457825},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.4333500266075134},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3863968551158905},{"id":"https://openalex.org/keywords/linguistics","display_name":"Linguistics","score":0.07524815201759338},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.06284719705581665}],"concepts":[{"id":"https://openalex.org/C198800484","wikidata":"https://www.wikidata.org/wiki/Q1503423","display_name":"Quasigroup","level":2,"score":0.946333646774292},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7337755560874939},{"id":"https://openalex.org/C43929256","wikidata":"https://www.wikidata.org/wiki/Q270513","display_name":"Catalan number","level":2,"score":0.6419904828071594},{"id":"https://openalex.org/C156340839","wikidata":"https://www.wikidata.org/wiki/Q2704791","display_name":"Enumeration","level":2,"score":0.5542621612548828},{"id":"https://openalex.org/C164105321","wikidata":"https://www.wikidata.org/wiki/Q7026","display_name":"Catalan","level":2,"score":0.5397335886955261},{"id":"https://openalex.org/C183222429","wikidata":"https://www.wikidata.org/wiki/Q42278","display_name":"Magma","level":3,"score":0.5208585858345032},{"id":"https://openalex.org/C98371066","wikidata":"https://www.wikidata.org/wiki/Q230848","display_name":"Euclidean algorithm","level":2,"score":0.5061778426170349},{"id":"https://openalex.org/C90805587","wikidata":"https://www.wikidata.org/wiki/Q10944557","display_name":"Word (group theory)","level":2,"score":0.48444512486457825},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.4333500266075134},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3863968551158905},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.07524815201759338},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.06284719705581665},{"id":"https://openalex.org/C127313418","wikidata":"https://www.wikidata.org/wiki/Q1069","display_name":"Geology","level":0,"score":0.0},{"id":"https://openalex.org/C120806208","wikidata":"https://www.wikidata.org/wiki/Q8072","display_name":"Volcano","level":2,"score":0.0},{"id":"https://openalex.org/C165205528","wikidata":"https://www.wikidata.org/wiki/Q83371","display_name":"Seismology","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1142/s0218196720500496","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218196720500496","pdf_url":null,"source":{"id":"https://openalex.org/S23260266","display_name":"International Journal of Algebra and Computation","issn_l":"0218-1967","issn":["0218-1967","1793-6500"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Algebra and Computation","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1910.09749","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1910.09749","pdf_url":"https://arxiv.org/pdf/1910.09749","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:2981369365","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1910.09749.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.1910.09749","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1910.09749","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"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1910.09749","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1910.09749","pdf_url":"https://arxiv.org/pdf/1910.09749","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":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2981369365.pdf","grobid_xml":"https://content.openalex.org/works/W2981369365.grobid-xml"},"referenced_works_count":13,"referenced_works":["https://openalex.org/W134303006","https://openalex.org/W186403072","https://openalex.org/W629021482","https://openalex.org/W656285810","https://openalex.org/W1996440833","https://openalex.org/W2092435222","https://openalex.org/W2133452007","https://openalex.org/W2138549733","https://openalex.org/W2340541605","https://openalex.org/W2505626532","https://openalex.org/W2752853835","https://openalex.org/W2804219209","https://openalex.org/W4246576549"],"related_works":["https://openalex.org/W2890145453","https://openalex.org/W1601772464","https://openalex.org/W271352609","https://openalex.org/W2003889772","https://openalex.org/W1614407559","https://openalex.org/W2561769070","https://openalex.org/W2736542075","https://openalex.org/W3165840172","https://openalex.org/W3099884582","https://openalex.org/W2797815838","https://openalex.org/W2955814494","https://openalex.org/W144207924","https://openalex.org/W2025761369","https://openalex.org/W3102088978","https://openalex.org/W2508096977","https://openalex.org/W2762976652","https://openalex.org/W3094388259","https://openalex.org/W2120533397","https://openalex.org/W193463686","https://openalex.org/W2963980262"],"abstract_inverted_index":{"This":[0],"paper":[1,51],"counts":[2],"the":[3,22,25,36,41,58,63,84,88,101,106,124,128],"number":[4,16,39,137],"of":[5,9,17,21,49,78,87,100,110,127,138],"reduced":[6],"quasigroup":[7,43,111,117],"words":[8,29,133],"a":[10,14,31,120,135],"particular":[11],"length":[12],"in":[13,30,134],"certain":[15],"generators.":[18,139],"Taking":[19],"account":[20],"relationship":[23],"with":[24],"Catalan":[26],"numbers,":[27,60,90],"counting":[28],"free":[32,42],"magma,":[33],"we":[34],"introduce":[35],"term":[37],"peri-Catalan":[38,59,89,129],"for":[40,57,83,131],"word":[44],"counts.":[45],"The":[46,66],"main":[47],"result":[48],"this":[50],"is":[52,105],"an":[53],"exact":[54],"recursive":[55],"formula":[56],"structured":[61],"by":[62,92],"Euclidean":[64,67],"Algorithm.":[65],"Algorithm":[68],"structure":[69],"does":[70],"not":[71],"readily":[72],"lend":[73],"itself":[74],"to":[75],"standard":[76],"techniques":[77],"asymptotic":[79,85,103,108,125],"analysis.":[80],"However,":[81],"conjectures":[82],"behavior":[86,104,126],"substantiated":[91],"numerical":[93],"data,":[94],"are":[95],"presented.":[96],"A":[97],"remarkable":[98],"aspect":[99],"observed":[102],"so-called":[107],"irrelevance":[109],"identities,":[112],"whereby":[113],"cancelation":[114],"resulting":[115],"from":[116],"identities":[118],"has":[119],"negligible":[121],"effect":[122],"on":[123],"numbers":[130],"long":[132],"large":[136]},"counts_by_year":[{"year":2019,"cited_by_count":1}],"updated_date":"2026-02-09T09:26:11.010843","created_date":"2025-10-10T00:00:00"}
