{"id":"https://openalex.org/W7204235360","doi":"https://doi.org/10.48550/arxiv.2608.24476","title":"Iterating the Lehmer code on inversion sequences: Catalan fixed points and finite stabilization","display_name":"Iterating the Lehmer code on inversion sequences: Catalan fixed points and finite stabilization","publication_year":2026,"publication_date":"2026-08-25","ids":{"openalex":"https://openalex.org/W7204235360","doi":"https://doi.org/10.48550/arxiv.2608.24476"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.24476","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24476","pdf_url":"https://arxiv.org/pdf/2608.24476","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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/2608.24476","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5038818656","display_name":"Julian D. Allagan","orcid":"https://orcid.org/0000-0003-0275-4239","observed_orcids":["https://orcid.org/0000-0003-0275-4239"]},"institutions":[{"id":"https://openalex.org/I22407884","display_name":"University of Maryland Eastern Shore","ror":"https://ror.org/006cymg18","country_code":"US","type":"education","lineage":["https://openalex.org/I22407884"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Allagan, Julian","raw_affiliation_strings":["Department of Mathematics , University of Maryland Eastern Shore , Princess Anne , MD 21853 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , University of Maryland Eastern Shore , Princess Anne , MD 21853 , USA","institution_ids":["https://openalex.org/I22407884"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5003535398","display_name":"Shanzhen Gao","orcid":"https://orcid.org/0000-0002-3856-2530","observed_orcids":["https://orcid.org/0000-0002-3856-2530"]},"institutions":[{"id":"https://openalex.org/I11786554","display_name":"Virginia State University","ror":"https://ror.org/04esvpn06","country_code":"US","type":"education","lineage":["https://openalex.org/I11786554"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Gao, Shanzhen","raw_affiliation_strings":["Department of Computer Information Systems , Virginia State University , Petersburg , VA 23806 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Information Systems , Virginia State University , Petersburg , VA 23806 , USA","institution_ids":["https://openalex.org/I11786554"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5080280861","display_name":"Benjamin Testart","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en sciences et technologies du num\u00e9rique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"government","lineage":["https://openalex.org/I1326498283"]},{"id":"https://openalex.org/I4210121838","display_name":"Laboratoire Lorrain de Recherche en Informatique et ses Applications","ror":"https://ror.org/02vnf0c38","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I277688954","https://openalex.org/I4210107720","https://openalex.org/I4210121838","https://openalex.org/I4210159245","https://openalex.org/I90183372"]},{"id":"https://openalex.org/I90183372","display_name":"Universit\u00e9 de Lorraine","ror":"https://ror.org/04vfs2w97","country_code":"FR","type":"education","lineage":["https://openalex.org/I90183372"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Testart, Benjamin","raw_affiliation_strings":["Universit\u00e9 de Lorraine , CNRS , Inria , LORIA , F-54000 Nancy , France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Universit\u00e9 de Lorraine , CNRS , Inria , LORIA , F-54000 Nancy , France","institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I4210121838","https://openalex.org/I90183372"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":6,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":1.0,"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":1.0,"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":0.0,"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/T13500","display_name":"Probability and Statistical Research","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/inversion-sequences","display_name":"inversion sequences","score":0.9980000257492065},{"id":"https://openalex.org/keywords/permutation-patterns","display_name":"permutation patterns","score":0.4880000054836273},{"id":"https://openalex.org/keywords/pattern-avoidance","display_name":"pattern avoidance","score":0.777999997138977},{"id":"https://openalex.org/keywords/fixed-points","display_name":"fixed points","score":0.7039999961853027},{"id":"https://openalex.org/keywords/catalan-numbers","display_name":"Catalan numbers","score":0.9459999799728394},{"id":"https://openalex.org/keywords/dyck-paths","display_name":"Dyck paths","score":0.9599999785423279},{"id":"https://openalex.org/keywords/bijections","display_name":"bijections","score":0.7409999966621399},{"id":"https://openalex.org/keywords/time-to-stabilization","display_name":"stabilization time","score":0.7509999871253967}],"concepts":[{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.8935999870300293},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8166999816894531},{"id":"https://openalex.org/C1893757","wikidata":"https://www.wikidata.org/wiki/Q3653001","display_name":"Inversion (geology)","level":3,"score":0.776199996471405},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.5568000078201294},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5260999798774719},{"id":"https://openalex.org/C43929256","wikidata":"https://www.wikidata.org/wiki/Q270513","display_name":"Catalan number","level":2,"score":0.42899999022483826},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.41780000925064087},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.40779998898506165},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.3910999894142151},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.37959998846054077},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.375900000333786},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.3562000095844269},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.34529998898506165},{"id":"https://openalex.org/C61994974","wikidata":"https://www.wikidata.org/wiki/Q2297602","display_name":"Integer sequence","level":3,"score":0.30660000443458557},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.30469998717308044},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.30399999022483826},{"id":"https://openalex.org/C88626702","wikidata":"https://www.wikidata.org/wiki/Q1128903","display_name":"Continuation","level":2,"score":0.27489998936653137},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.27000001072883606},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.25699999928474426}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.24476","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24476","pdf_url":"https://arxiv.org/pdf/2608.24476","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.24476","doi":"https://doi.org/10.48550/arxiv.2608.24476","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.24476","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.24476","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24476","pdf_url":"https://arxiv.org/pdf/2608.24476","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204235360.pdf","grobid_xml":"https://content.openalex.org/works/W7204235360.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W1988028638","https://openalex.org/W2887852849","https://openalex.org/W7221375054","https://openalex.org/W4206564921","https://openalex.org/W2953060988","https://openalex.org/W3037553967","https://openalex.org/W4415030244","https://openalex.org/W7172402205","https://openalex.org/W4362639815","https://openalex.org/W4391127817"],"abstract_inverted_index":{"We":[0,64,100],"study":[1],"an":[2,43,93],"operator":[3,26],"$\u0398$":[4,50,70],"on":[5],"finite":[6,122],"integer":[7],"sequences,":[8,127],"where":[9],"$\u0398(\u03c3)_i$":[10],"counts":[11],"the":[12,15,31,39,47,66,74,88,104,132,139],"entries":[13],"to":[14,51],"left":[16],"of":[17,30,49,53,61,69,73],"$\u03c3_i$":[18],"that":[19,83,103,138],"are":[20,85,111],"strictly":[21],"smaller":[22],"than":[23],"$\u03c3_i$.":[24],"This":[25],"is":[27,42,55,109,143],"a":[28,56,79,129],"variant":[29],"so-called":[32],"Lehmer":[33],"code.":[34],"For":[35],"every":[36],"sequence":[37],"$\u03c3$,":[38],"image":[40],"$\u0398(\u03c3)$":[41],"inversion":[44,59,126],"sequence,":[45],"and":[46,91,117,136],"restriction":[48],"permutations":[52],"$[0,n-1]$":[54],"bijection":[57,96],"onto":[58],"sequences":[60,105],"length":[62],"$n$.":[63],"characterize":[65],"fixed":[67,110],"points":[68],"by":[71,87],"avoidance":[72],"pattern":[75],"$101$":[76,116],"together":[77],"with":[78,97],"saturation":[80],"condition,":[81],"prove":[82,121],"they":[84],"counted":[86],"Catalan":[89],"numbers,":[90],"give":[92],"explicit":[94],"recursive":[95],"Dyck":[98],"paths.":[99],"also":[101],"show":[102,137],"whose":[106],"first":[107],"$\u0398$-image":[108],"precisely":[112],"those":[113],"avoiding":[114],"both":[115],"$201$.":[118],"Finally,":[119],"we":[120],"stabilization":[123,134,141],"for":[124],"all":[125],"exhibit":[128],"family":[130],"attaining":[131],"maximal":[133],"time,":[135],"second":[140],"level":[142],"not":[144],"closed":[145],"under":[146],"classical":[147],"patterns.":[148]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-08-27T00:00:00"}
