{"id":"https://openalex.org/W7220377663","doi":"https://doi.org/10.48550/arxiv.2610.02756","title":"Motzkin Numbers Count 2-Stack-Sortable Permutations Ending in Their Least Entry","display_name":"Motzkin Numbers Count 2-Stack-Sortable Permutations Ending in Their Least Entry","publication_year":2026,"publication_date":"2026-10-02","ids":{"openalex":"https://openalex.org/W7220377663","doi":"https://doi.org/10.48550/arxiv.2610.02756"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.02756","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02756","pdf_url":"https://arxiv.org/pdf/2610.02756","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/2610.02756","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5087772695","display_name":"Ryota Inagaki","orcid":"https://orcid.org/0000-0002-2015-8492","observed_orcids":["https://orcid.org/0000-0002-2015-8492"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Inagaki, Ryota","raw_affiliation_strings":["Minnetonka High School 18301 MN -7 , Minnetonka , Minnesota , U.S.A. 55345"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Minnetonka High School 18301 MN -7 , Minnetonka , Minnesota , U.S.A. 55345","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5035392658","display_name":"Michael Luo","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Luo, Michael","raw_affiliation_strings":["Minnetonka High School 18301 MN -7 , Minnetonka , Minnesota , U.S.A. 55345"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Minnetonka High School 18301 MN -7 , Minnetonka , Minnesota , U.S.A. 55345","institution_ids":[]}]}],"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":1,"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/T11428","display_name":"Advanced Mathematical Identities","score":0.0,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/motzkin-numbers","display_name":"Motzkin numbers","score":1.0},{"id":"https://openalex.org/keywords/permutation-enumeration","display_name":"permutation enumeration","score":0.6940000057220459},{"id":"https://openalex.org/keywords/bijections","display_name":"bijections","score":0.8399999737739563},{"id":"https://openalex.org/keywords/standard-young-tableaux","display_name":"standard Young tableaux","score":0.9430000185966492},{"id":"https://openalex.org/keywords/permutation-sorting","display_name":"permutation sorting","score":0.453000009059906}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8543999791145325},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.8141999840736389},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.796500027179718},{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.7705000042915344},{"id":"https://openalex.org/C2777045944","wikidata":"https://www.wikidata.org/wiki/Q12170198","display_name":"Zh\u00e0ng","level":3,"score":0.40070000290870667},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3926999866962433},{"id":"https://openalex.org/C200288055","wikidata":"https://www.wikidata.org/wiki/Q2621792","display_name":"Element (criminal law)","level":2,"score":0.34779998660087585},{"id":"https://openalex.org/C21308566","wikidata":"https://www.wikidata.org/wiki/Q7169365","display_name":"Permutation (music)","level":2,"score":0.3237000107765198},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.3109999895095825},{"id":"https://openalex.org/C40231798","wikidata":"https://www.wikidata.org/wiki/Q1333743","display_name":"Composition (language)","level":2,"score":0.3043999969959259},{"id":"https://openalex.org/C165368118","wikidata":"https://www.wikidata.org/wiki/Q2166280","display_name":"Young tableau","level":2,"score":0.2802000045776367},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.2612999975681305}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.02756","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02756","pdf_url":"https://arxiv.org/pdf/2610.02756","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.2610.02756","doi":"https://doi.org/10.48550/arxiv.2610.02756","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.02756","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:2610.02756","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02756","pdf_url":"https://arxiv.org/pdf/2610.02756","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":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220377663.pdf","grobid_xml":"https://content.openalex.org/works/W7220377663.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W7154212160"],"related_works":["https://openalex.org/W7096636033","https://openalex.org/W2069405390","https://openalex.org/W7154212160","https://openalex.org/W2597896914","https://openalex.org/W3134398167","https://openalex.org/W2258801652","https://openalex.org/W7223208658","https://openalex.org/W1619465886","https://openalex.org/W1938446813","https://openalex.org/W1724407287"],"abstract_inverted_index":{"We":[0,53],"prove":[1],"the":[2,14,25,81],"following":[3],"conjecture":[4],"of":[5,16,19,48,63,69],"Zhang":[6],"(arXiv:2604.10779,":[7],"Conjecture":[8],"6.1):":[9],"for":[10],"$n":[11],"\\geq":[12],"0$,":[13],"number":[15,62],"$2$-stack-sortable":[17,37],"permutations":[18,38],"$\\{0,1,\\dots,n\\}$":[20],"ending":[21,39],"in":[22,40],"$0$":[23],"is":[24,33],"$n$th":[26],"Motzkin":[27,82],"number.":[28],"By":[29],"Zhang's":[30],"result,":[31],"there":[32,58],"a":[34],"bijection":[35],"between":[36],"their":[41],"least":[42],"element":[43],"and":[44,65],"standard":[45,66],"composition":[46],"tableaux":[47,68],"width":[49,70],"at":[50,71],"most":[51,72],"$2$.":[52],"then":[54],"show":[55],"bijectively":[56],"that":[57],"are":[59,75],"an":[60],"equal":[61],"these":[64],"Young":[67],"$3$,":[73],"which":[74],"known":[76],"to":[77],"be":[78],"counted":[79],"by":[80],"numbers.":[83]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-06T00:00:00"}
