{"id":"https://openalex.org/W7214237291","doi":"https://doi.org/10.48550/arxiv.2609.28059","title":"Abelian maximal pattern complexity and extremal words","display_name":"Abelian maximal pattern complexity and extremal words","publication_year":2026,"publication_date":"2026-09-23","ids":{"openalex":"https://openalex.org/W7214237291","doi":"https://doi.org/10.48550/arxiv.2609.28059"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.28059","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.28059","pdf_url":"https://arxiv.org/pdf/2609.28059","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.28059","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5103169944","display_name":"Qingcheng Zeng","orcid":"https://orcid.org/0009-0001-5196-0939","observed_orcids":["https://orcid.org/0009-0001-5196-0939"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zeng, Qingcheng","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5031868439","display_name":"Yumei Xue","orcid":"https://orcid.org/0000-0002-2103-0052","observed_orcids":["https://orcid.org/0000-0002-2103-0052"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xue, Yumei","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5152756116","display_name":"Cheng Zeng","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zeng, Cheng","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":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9794999957084656,"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"}},"topics":[{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9794999957084656,"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.0034000000450760126,"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"}},{"id":"https://openalex.org/T11269","display_name":"Algorithms and Data Compression","score":0.003000000026077032,"subfield":{"id":"https://openalex.org/subfields/2614","display_name":"Theoretical Computer Science"},"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/infinite-words","display_name":"infinite words","score":0.6209999918937683},{"id":"https://openalex.org/keywords/characteristic-words","display_name":"characteristic words","score":0.5260000228881836},{"id":"https://openalex.org/keywords/bipartite-graphs","display_name":"bipartite graphs","score":0.7480000257492065},{"id":"https://openalex.org/keywords/lower-bounds","display_name":"lower bounds","score":0.4269999861717224},{"id":"https://openalex.org/keywords/finite-alphabet","display_name":"finite alphabet","score":0.3779999911785126}],"concepts":[{"id":"https://openalex.org/C136170076","wikidata":"https://www.wikidata.org/wiki/Q181296","display_name":"Abelian group","level":2,"score":0.8385000228881836},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8330000042915344},{"id":"https://openalex.org/C104247578","wikidata":"https://www.wikidata.org/wiki/Q4779368","display_name":"Aperiodic graph","level":2,"score":0.8119999766349792},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7457000017166138},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5454000234603882},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.5397999882698059},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.49059998989105225},{"id":"https://openalex.org/C90805587","wikidata":"https://www.wikidata.org/wiki/Q10944557","display_name":"Word (group theory)","level":2,"score":0.4869999885559082},{"id":"https://openalex.org/C200681373","wikidata":"https://www.wikidata.org/wiki/Q1260803","display_name":"Rank of an abelian group","level":4,"score":0.45910000801086426},{"id":"https://openalex.org/C197657726","wikidata":"https://www.wikidata.org/wiki/Q174733","display_name":"Bipartite graph","level":3,"score":0.4422000050544739},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.42559999227523804},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.39399999380111694},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.37929999828338623},{"id":"https://openalex.org/C81008192","wikidata":"https://www.wikidata.org/wiki/Q1017231","display_name":"Elementary abelian group","level":3,"score":0.37700000405311584},{"id":"https://openalex.org/C186921422","wikidata":"https://www.wikidata.org/wiki/Q318737","display_name":"Abelian category","level":3,"score":0.33309999108314514},{"id":"https://openalex.org/C173734053","wikidata":"https://www.wikidata.org/wiki/Q47577","display_name":"Fibonacci number","level":2,"score":0.3303999900817871},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.3287000060081482},{"id":"https://openalex.org/C164804908","wikidata":"https://www.wikidata.org/wiki/Q2985058","display_name":"Combinatorics on words","level":3,"score":0.3142000138759613},{"id":"https://openalex.org/C193319292","wikidata":"https://www.wikidata.org/wiki/Q272172","display_name":"Hamming distance","level":2,"score":0.3084000051021576},{"id":"https://openalex.org/C7655956","wikidata":"https://www.wikidata.org/wiki/Q5282259","display_name":"Disjoint union (topology)","level":2,"score":0.265500009059906}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.28059","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.28059","pdf_url":"https://arxiv.org/pdf/2609.28059","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.28059","doi":"https://doi.org/10.48550/arxiv.2609.28059","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.28059","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.28059","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.28059","pdf_url":"https://arxiv.org/pdf/2609.28059","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320324174","display_name":"Natural Science Foundation of Shandong Province","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214237291.pdf","grobid_xml":"https://content.openalex.org/works/W7214237291.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W4414561363"],"related_works":["https://openalex.org/W4300805396","https://openalex.org/W2005958627","https://openalex.org/W2059751203","https://openalex.org/W7204116881","https://openalex.org/W4417085104","https://openalex.org/W7204659526","https://openalex.org/W2008695999","https://openalex.org/W2113070578","https://openalex.org/W2012233760","https://openalex.org/W3214596242"],"abstract_inverted_index":{"We":[0,115,144],"study":[1],"the":[2,26,104,136,163],"Abelian":[3,106,120,131,150],"maximal":[4,107,132],"pattern":[5,108,121,133,151],"complexity":[6,109,134],"$p_{\u03b1}^{\\ast":[7,30],"\\mathrm{ab}}(k)$":[8],"of":[9,38,166,184],"infinite":[10,54,168],"words":[11,18,40,73],"over":[12,74],"finite":[13],"alphabets.":[14],"For":[15,71],"recurrent":[16],"aperiodic":[17,58,72],"using":[19],"exactly":[20,201],"$\\ell\\geq":[21,75],"2$":[22,76],"letters,":[23,77],"we":[24,50,82],"prove":[25,51,84],"sharp":[27],"lower":[28],"bound":[29],"\\mathrm{ab}}(k)\\geq":[31,87],"k+\\ell-1$":[32],"and":[33,60,154],"give":[34],"a":[35,98,117,147,176,191,197],"structural":[36],"characterization":[37],"those":[39],"attaining":[41],"equality":[42],"for":[43,66,91,125],"every":[44,67,126],"$k\\geq":[45],"1$.":[46],"Without":[47],"assuming":[48],"recurrence,":[49],"that":[52,85,103,146],"an":[53,119,167],"word":[55,118,123,148,165],"$\u03b1$":[56],"is":[57,113,135,149,162,175],"if":[59,62,153],"only":[61,155],"$\\binom{p_{\u03b1}^{\\ast":[63],"\\mathrm{ab}}(k)}{2}\\geq":[64],"k$":[65],"positive":[68,93,127,138],"integer":[69,128,139],"$k$.":[70],"each":[78],"occurring":[79],"infinitely":[80],"often,":[81],"further":[83],"$p_\u03b1^{\\ast":[86],"m$":[88],"whenever":[89],"$\\binom{m}{2}\\leq(\\ell-1)(k-\\ell+2)$,":[90],"all":[92],"integers":[94],"$m,k$.":[95],"Together":[96],"with":[97,190],"matching":[99],"construction,":[100],"this":[101,111],"shows":[102],"minimum":[105],"in":[110],"class":[112],"$\\sqrt{2(\\ell-1)k}+O_{\\ell}(1)$.":[114],"call":[116],"Sturmian":[122,152],"if,":[124,156],"$k$,":[129],"its":[130,159,187],"least":[137],"$m$":[140],"satisfying":[141],"$\\binom{m}{2}\\geq":[142],"k$.":[143],"show":[145],"after":[157],"relabeling":[158],"alphabet,":[160],"it":[161],"characteristic":[164],"set":[169],"$E\\subset\\mathbb{N}_0$":[170],"whose":[171],"bipartite":[172],"incidence":[173],"graph":[174,179],"forest.":[177],"This":[178],"has":[180],"two":[181],"disjoint":[182],"copies":[183],"$\\mathbb{N}_0$":[185],"as":[186],"vertex":[188,193,199],"classes,":[189],"left":[192],"$r$":[194],"adjacent":[195],"to":[196],"right":[198],"$s$":[200],"when":[202],"$r+s\\in":[203],"E$.":[204]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-09-25T00:00:00"}
