{"id":"https://openalex.org/W1562379951","doi":"https://doi.org/10.37236/937","title":"The Number of [Old-Time] Basketball Games with Final Score $n$:$n$ where the Home Team was Never Losing but also Never Ahead by More Than $w$ Points","display_name":"The Number of [Old-Time] Basketball Games with Final Score $n$:$n$ where the Home Team was Never Losing but also Never Ahead by More Than $w$ Points","publication_year":2007,"publication_date":"2007-01-29","ids":{"openalex":"https://openalex.org/W1562379951","doi":"https://doi.org/10.37236/937","mag":"1562379951"},"language":"en","primary_location":{"id":"doi:10.37236/937","is_oa":true,"landing_page_url":"https://doi.org/10.37236/937","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v14i1r19/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":null,"license_id":null,"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/v14i1r19/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5028829574","display_name":"Arvind Ayyer","orcid":"https://orcid.org/0000-0002-9271-2421"},"institutions":[{"id":"https://openalex.org/I4210109165","display_name":"Environmental and Occupational Health Sciences Institute","ror":"https://ror.org/01vta4r13","country_code":"US","type":"education","lineage":["https://openalex.org/I4210109165"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Arvind Ayyer","raw_affiliation_strings":["Department of Physics 136 Frelinghuysen Rd Piscataway, NJ 08854","Department of Physics"],"affiliations":[{"raw_affiliation_string":"Department of Physics 136 Frelinghuysen Rd Piscataway, NJ 08854","institution_ids":["https://openalex.org/I4210109165"]},{"raw_affiliation_string":"Department of Physics","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5058881306","display_name":"Doron Zeilberger","orcid":null},"institutions":[{"id":"https://openalex.org/I4210109165","display_name":"Environmental and Occupational Health Sciences Institute","ror":"https://ror.org/01vta4r13","country_code":"US","type":"education","lineage":["https://openalex.org/I4210109165"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Doron Zeilberger","raw_affiliation_strings":["Department of Mathematics 110 Frelinghuysen Rd Piscataway, NJ 08854","School of Arts and Sciences, Mathematics"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics 110 Frelinghuysen Rd Piscataway, NJ 08854","institution_ids":["https://openalex.org/I4210109165"]},{"raw_affiliation_string":"School of Arts and Sciences, Mathematics","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5028829574"],"corresponding_institution_ids":["https://openalex.org/I4210109165"],"apc_list":null,"apc_paid":null,"fwci":6.9586,"has_fulltext":true,"cited_by_count":41,"citation_normalized_percentile":{"value":0.97370317,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"14","issue":"1","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":0.9952999949455261,"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":0.9952999949455261,"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9940999746322632,"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":0.9926000237464905,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8643423318862915},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7264114022254944},{"id":"https://openalex.org/keywords/recurrence-relation","display_name":"Recurrence relation","score":0.6912434101104736},{"id":"https://openalex.org/keywords/basketball","display_name":"Basketball","score":0.6784671545028687},{"id":"https://openalex.org/keywords/relation","display_name":"Relation (database)","score":0.5270707011222839},{"id":"https://openalex.org/keywords/generating-function","display_name":"Generating function","score":0.511181652545929},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.49510660767555237},{"id":"https://openalex.org/keywords/square","display_name":"Square (algebra)","score":0.44820550084114075},{"id":"https://openalex.org/keywords/lattice","display_name":"Lattice (music)","score":0.42913544178009033},{"id":"https://openalex.org/keywords/square-lattice","display_name":"Square lattice","score":0.41371724009513855},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3536415994167328},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.07479357719421387},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.06713658571243286},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.05366313457489014}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8643423318862915},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7264114022254944},{"id":"https://openalex.org/C201292218","wikidata":"https://www.wikidata.org/wiki/Q740970","display_name":"Recurrence relation","level":2,"score":0.6912434101104736},{"id":"https://openalex.org/C103189561","wikidata":"https://www.wikidata.org/wiki/Q5372","display_name":"Basketball","level":2,"score":0.6784671545028687},{"id":"https://openalex.org/C25343380","wikidata":"https://www.wikidata.org/wiki/Q277521","display_name":"Relation (database)","level":2,"score":0.5270707011222839},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.511181652545929},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.49510660767555237},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.44820550084114075},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.42913544178009033},{"id":"https://openalex.org/C2777620828","wikidata":"https://www.wikidata.org/wiki/Q7582067","display_name":"Square lattice","level":3,"score":0.41371724009513855},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3536415994167328},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.07479357719421387},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.06713658571243286},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.05366313457489014},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C95457728","wikidata":"https://www.wikidata.org/wiki/Q309","display_name":"History","level":0,"score":0.0},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C166957645","wikidata":"https://www.wikidata.org/wiki/Q23498","display_name":"Archaeology","level":1,"score":0.0},{"id":"https://openalex.org/C51329190","wikidata":"https://www.wikidata.org/wiki/Q1076349","display_name":"Ising model","level":2,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.37236/937","is_oa":true,"landing_page_url":"https://doi.org/10.37236/937","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v14i1r19/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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.178.5368","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.178.5368","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.combinatorics.org/Volume_14/PDF/v14i1r19.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.234.3247","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.234.3247","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/math/0610734v1.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.234.4113","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.234.4113","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/math/0610734v2.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/937","is_oa":true,"landing_page_url":"https://doi.org/10.37236/937","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v14i1r19/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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Quality Education","score":0.7200000286102295,"id":"https://metadata.un.org/sdg/4"}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1562379951.pdf","grobid_xml":"https://content.openalex.org/works/W1562379951.grobid-xml"},"referenced_works_count":7,"referenced_works":["https://openalex.org/W1504241636","https://openalex.org/W1966738703","https://openalex.org/W2026690496","https://openalex.org/W2063672317","https://openalex.org/W3099024983","https://openalex.org/W4388317767","https://openalex.org/W6600283948"],"related_works":["https://openalex.org/W2551082991","https://openalex.org/W2049139273","https://openalex.org/W2071588082","https://openalex.org/W203290572","https://openalex.org/W2418411990","https://openalex.org/W2189105445","https://openalex.org/W1536046884","https://openalex.org/W3166262075","https://openalex.org/W4318426929","https://openalex.org/W2953049497"],"abstract_inverted_index":{"We":[0],"show":[1],"that":[2,47],"the":[3,9,14,29],"generating":[4],"function":[5],"(in":[6],"$n$)":[7],"for":[8],"number":[10],"of":[11],"walks":[12],"on":[13],"square":[15],"lattice":[16],"with":[17],"steps":[18],"$(1,1),":[19],"(1,-1),":[20],"(2,2)$":[21],"and":[22,52],"$(2,-2)$":[23],"from":[24],"$(0,0)$":[25],"to":[26],"$(2n,0)$":[27],"in":[28,45],"region":[30],"$0":[31],"\\leq":[32,34],"y":[33],"w$":[35],"satisfies":[36],"a":[37,55],"very":[38],"special":[39],"fifth":[40],"order":[41],"nonlinear":[42],"recurrence":[43,57],"relation":[44],"$w$":[46],"implies":[48],"both":[49],"its":[50],"numerator":[51],"denominator":[53],"satisfy":[54],"linear":[56],"relation.":[58]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":6}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}