{"id":"https://openalex.org/W7119819077","doi":"https://doi.org/10.37236/13115","title":"Combinatorial Enumeration of Lattice Paths by Flaws with Respect to a Linear Boundary of Rational Slope","display_name":"Combinatorial Enumeration of Lattice Paths by Flaws with Respect to a Linear Boundary of Rational Slope","publication_year":2026,"publication_date":"2026-01-09","ids":{"openalex":"https://openalex.org/W7119819077","doi":"https://doi.org/10.37236/13115"},"language":null,"primary_location":{"id":"doi:10.37236/13115","is_oa":true,"landing_page_url":"https://doi.org/10.37236/13115","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v33i1p3/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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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/v33i1p3/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5099172287","display_name":"Federico Firoozi","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Federico Firoozi","raw_affiliation_strings":["Department of Mathematics and Statistics"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5122550202","display_name":"Jonathan Jedwab","orcid":null},"institutions":[{"id":"https://openalex.org/I18014758","display_name":"Simon Fraser University","ror":"https://ror.org/0213rcc28","country_code":"CA","type":"education","lineage":["https://openalex.org/I18014758"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Jonathan Jedwab","raw_affiliation_strings":["Simon Fraser University"],"affiliations":[{"raw_affiliation_string":"Simon Fraser University","institution_ids":["https://openalex.org/I18014758"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5067538348","display_name":"Amarpreet Rattan","orcid":null},"institutions":[{"id":"https://openalex.org/I530967","display_name":"Toronto Metropolitan University","ror":"https://ror.org/05g13zd79","country_code":"CA","type":"education","lineage":["https://openalex.org/I530967"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Amarpreet Rattan","raw_affiliation_strings":["Department of Mathematics"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics","institution_ids":["https://openalex.org/I530967"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5099172287"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.12034161,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"33","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":0.31220000982284546,"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":0.31220000982284546,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.2946999967098236,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.08049999922513962,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/enumeration","display_name":"Enumeration","score":0.782800018787384},{"id":"https://openalex.org/keywords/bijection","display_name":"Bijection","score":0.7412999868392944},{"id":"https://openalex.org/keywords/coprime-integers","display_name":"Coprime integers","score":0.6620000004768372},{"id":"https://openalex.org/keywords/lattice","display_name":"Lattice (music)","score":0.6312999725341797},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.5073999762535095},{"id":"https://openalex.org/keywords/recursion","display_name":"Recursion (computer science)","score":0.48590001463890076},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.4668000042438507},{"id":"https://openalex.org/keywords/fixed-point","display_name":"Fixed point","score":0.4219000041484833},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.3833000063896179}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7971000075340271},{"id":"https://openalex.org/C156340839","wikidata":"https://www.wikidata.org/wiki/Q2704791","display_name":"Enumeration","level":2,"score":0.782800018787384},{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.7412999868392944},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.6620000004768372},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.6312999725341797},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6301000118255615},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.5073999762535095},{"id":"https://openalex.org/C168773036","wikidata":"https://www.wikidata.org/wiki/Q264164","display_name":"Recursion (computer science)","level":2,"score":0.48590001463890076},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.4668000042438507},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.4219000041484833},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4027999937534332},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.3833000063896179},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3684999942779541},{"id":"https://openalex.org/C95611797","wikidata":"https://www.wikidata.org/wiki/Q17502105","display_name":"Infimum and supremum","level":2,"score":0.34439998865127563},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.34209999442100525},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.32679998874664307},{"id":"https://openalex.org/C2776291640","wikidata":"https://www.wikidata.org/wiki/Q2912517","display_name":"Value (mathematics)","level":2,"score":0.29829999804496765},{"id":"https://openalex.org/C142885752","wikidata":"https://www.wikidata.org/wiki/Q1252145","display_name":"Integer lattice","level":3,"score":0.28839999437332153},{"id":"https://openalex.org/C34718186","wikidata":"https://www.wikidata.org/wiki/Q209875","display_name":"Binomial coefficient","level":2,"score":0.2858999967575073},{"id":"https://openalex.org/C201292218","wikidata":"https://www.wikidata.org/wiki/Q740970","display_name":"Recurrence relation","level":2,"score":0.27390000224113464},{"id":"https://openalex.org/C164680029","wikidata":"https://www.wikidata.org/wiki/Q1055293","display_name":"Multiple","level":2,"score":0.27070000767707825},{"id":"https://openalex.org/C48659774","wikidata":"https://www.wikidata.org/wiki/Q4907197","display_name":"Bijection, injection and surjection","level":3,"score":0.2671000063419342},{"id":"https://openalex.org/C46875033","wikidata":"https://www.wikidata.org/wiki/Q1244890","display_name":"Rational number","level":2,"score":0.2583000063896179},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.25540000200271606}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/13115","is_oa":true,"landing_page_url":"https://doi.org/10.37236/13115","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v33i1p3/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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.37236/13115","is_oa":true,"landing_page_url":"https://doi.org/10.37236/13115","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v33i1p3/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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7119819077.pdf","grobid_xml":"https://content.openalex.org/works/W7119819077.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Let":[0],"$a,b$":[1],"be":[2],"fixed":[3],"positive":[4,9],"coprime":[5],"integers.":[6],"For":[7],"a":[8,67,75,121,156,176],"integer":[10],"$g$,":[11],"write":[12],"$W_k(g)$":[13,125],"for":[14,54,72,158,168,180,182],"the":[15,21,25,42,46,49,85,100,135],"set":[16,143],"of":[17,61,88,97,124,144,147],"lattice":[18,62],"paths":[19,63],"from":[20],"startpoint":[22,47],"$(0,0)$":[23],"to":[24,31,48,66,84,154,174],"endpoint":[26],"$(ga,gb)$":[27],"with":[28,64,129],"steps":[29],"restricted":[30],"$\\{(1,0),":[32],"(0,1)\\}$,":[33],"having":[34],"exactly":[35],"$k$":[36,56,184],"flaws":[37,73],"(lattice":[38],"points":[39],"lying":[40],"above":[41],"linear":[43,68],"boundary":[44,69],"connecting":[45],"endpoint).":[50],"We":[51,170],"determine":[52],"$|W_k(g)|$":[53,98,137,159,181],"all":[55,183],"and":[57,77,90,106,185],"$g$.":[58,186],"The":[59,92],"enumeration":[60],"respect":[65],"while":[70],"accounting":[71],"has":[74],"long":[76],"rich":[78],"history,":[79],"dating":[80],"back":[81],"at":[82],"least":[83],"1949":[86],"results":[87],"Chung":[89],"Feller.":[91],"only":[93],"previously":[94],"known":[95],"values":[96,146],"are":[99,189],"extremal":[101],"cases":[102],"$k":[103,107],"=":[104,108],"0$":[105],"g(a+b)-1$,":[109],"determined":[110],"by":[111,165],"Bizley":[112],"in":[113,127,150],"1954.":[114],"Our":[115,187],"main":[116],"combinatorial":[117],"result":[118,167],"is":[119,126,133,138,163],"that":[120,134],"certain":[122],"subset":[123],"bijection":[128],"$W_{k+1}(g)$.":[130],"One":[131],"consequence":[132],"value":[136],"constant":[139],"over":[140],"each":[141],"successive":[142],"$a+b$":[145],"$k$.":[148],"This":[149],"turn":[151],"allows":[152],"us":[153],"derive":[155],"recursion":[157,173],"whose":[160],"base":[161],"case":[162],"given":[164],"Bizley's":[166],"$k=0$.":[169],"solve":[171],"this":[172],"obtain":[175],"closed":[177],"form":[178],"expression":[179],"methods":[188],"purely":[190],"combinatorial.":[191]},"counts_by_year":[],"updated_date":"2026-03-12T06:13:28.667946","created_date":"2026-01-10T00:00:00"}
