{"id":"https://openalex.org/W1747715124","doi":"https://doi.org/10.37236/1914","title":"Splitting Numbers of Grids","display_name":"Splitting Numbers of Grids","publication_year":2005,"publication_date":"2005-04-13","ids":{"openalex":"https://openalex.org/W1747715124","doi":"https://doi.org/10.37236/1914","mag":"1747715124"},"language":"en","primary_location":{"id":"doi:10.37236/1914","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1914","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v12i1r17/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/v12i1r17/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5049969933","display_name":"Dwight Duffus","orcid":null},"institutions":[{"id":"https://openalex.org/I150468666","display_name":"Emory University","ror":"https://ror.org/03czfpz43","country_code":"US","type":"education","lineage":["https://openalex.org/I150468666"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Dwight Duffus","raw_affiliation_strings":["Mathematics and Computer Science Department Emory University Atlanta GA 30322 USA","Emory University"],"affiliations":[{"raw_affiliation_string":"Mathematics and Computer Science Department Emory University Atlanta GA 30322 USA","institution_ids":["https://openalex.org/I150468666"]},{"raw_affiliation_string":"Emory University","institution_ids":["https://openalex.org/I150468666"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5055969467","display_name":"Bill Sands","orcid":null},"institutions":[{"id":"https://openalex.org/I168635309","display_name":"University of Calgary","ror":"https://ror.org/03yjb2x39","country_code":"CA","type":"education","lineage":["https://openalex.org/I168635309"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Bill Sands","raw_affiliation_strings":["Mathematics and Statistics Department The University of Calgary Calgary AB T2N 1N4 CANADA","University of Calgary"],"affiliations":[{"raw_affiliation_string":"Mathematics and Statistics Department The University of Calgary Calgary AB T2N 1N4 CANADA","institution_ids":["https://openalex.org/I168635309"]},{"raw_affiliation_string":"University of Calgary","institution_ids":["https://openalex.org/I168635309"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5049969933"],"corresponding_institution_ids":["https://openalex.org/I150468666"],"apc_list":null,"apc_paid":null,"fwci":1.01,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.75792051,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"12","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9998999834060669,"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9998999834060669,"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/T11671","display_name":"Fuzzy and Soft Set Theory","score":0.9940999746322632,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9939000010490417,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/antichain","display_name":"Antichain","score":0.9570438861846924},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8354036808013916},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.785125195980072},{"id":"https://openalex.org/keywords/partially-ordered-set","display_name":"Partially ordered set","score":0.35900306701660156},{"id":"https://openalex.org/keywords/crystallography","display_name":"Crystallography","score":0.35130566358566284},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.07842963933944702}],"concepts":[{"id":"https://openalex.org/C136134351","wikidata":"https://www.wikidata.org/wiki/Q431937","display_name":"Antichain","level":3,"score":0.9570438861846924},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8354036808013916},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.785125195980072},{"id":"https://openalex.org/C180645754","wikidata":"https://www.wikidata.org/wiki/Q474715","display_name":"Partially ordered set","level":2,"score":0.35900306701660156},{"id":"https://openalex.org/C8010536","wikidata":"https://www.wikidata.org/wiki/Q160398","display_name":"Crystallography","level":1,"score":0.35130566358566284},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.07842963933944702}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.37236/1914","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1914","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v12i1r17/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:10.1.1.214.7086","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.214.7086","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_12/PDF/v12i1r17.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.379.8926","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.379.8926","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.emory.edu/technical-reports/techrep-00048.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/1914","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1914","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v12i1r17/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":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1747715124.pdf","grobid_xml":"https://content.openalex.org/works/W1747715124.grobid-xml"},"referenced_works_count":6,"referenced_works":["https://openalex.org/W1855706715","https://openalex.org/W2003755322","https://openalex.org/W2033523934","https://openalex.org/W2084042630","https://openalex.org/W2128978199","https://openalex.org/W2136516706"],"related_works":["https://openalex.org/W2013313833","https://openalex.org/W28336341","https://openalex.org/W2111205855","https://openalex.org/W2015816871","https://openalex.org/W1854642140","https://openalex.org/W2056881277","https://openalex.org/W2065333210","https://openalex.org/W1504624230","https://openalex.org/W3213974985","https://openalex.org/W4302107529"],"abstract_inverted_index":{"For":[0,28],"a":[1,5,29,68,96],"subset":[2],"$S$":[3],"of":[4,33,47,63,72,95,99,139],"finite":[6,73],"ordered":[7,74],"set":[8],"$P$,":[9,34],"let":[10,35],"$$S\\!\\uparrow\\;=\\{x\\in":[11],"P:x\\ge":[12],"s":[13,22],"\\hbox{":[14,23],"for":[15,24,67,126],"some":[16,25],"}s\\in":[17,26],"S\\}\\quad\\hbox{and}":[18],"\\quad":[19],"S\\!\\downarrow\\;=\\{x\\in":[20],"P:x\\le":[21],"S\\}.$$":[27],"maximal":[30,61,93],"antichain":[31,62,94],"$A$":[32],"$$s(A)=\\max_{A=U\\cup":[36],"D}{|U\\!\\uparrow|+|D\\!\\downarrow|\\over|P|}\\":[37],",$$":[38],"the":[39,86,137],"maximum":[40],"taken":[41],"over":[42],"all":[43,127,140],"partitions":[44],"$U\\cup":[45],"D$":[46],"$A$,":[48],"and":[49,108],"$$s_k(P)=\\min_{A\\in":[50],"{\\cal":[51,80],"A}(P),|A|=k}s(A)$$":[52],"where":[53,130],"we":[54,76,122],"assume":[55],"$P$":[56,98],"contains":[57,114],"at":[58,115],"least":[59,116],"one":[60],"$k$":[64,141],"elements.":[65,118],"Finally,":[66],"class":[69],"${\\cal":[70,100,131],"C}$":[71,101],"sets,":[75],"define":[77],"$$s_k({\\cal":[78],"C})=\\inf_{P\\in":[79],"C}}s_k(P).$$":[81],"Thus":[82],"$s_k({\\cal":[83,124],"C})$":[84],"is":[85,136],"greatest":[87],"proportion":[88],"$r$":[89],"satisfying:":[90],"every":[91],"$k$-element":[92],"member":[97],"can":[102],"be":[103],"\"split\"":[104],"into":[105],"sets":[106],"$U$":[107],"$D$":[109],"so":[110],"that":[111],"$U\\!\\uparrow\\cup\\;":[112],"D\\!\\downarrow$":[113],"$r|P|$":[117],"In":[119],"this":[120],"paper":[121],"determine":[123],"G}_k)$":[125],"$k\\ge":[128],"1$,":[129],"G}_k=\\{{\\bf":[132],"k}\\times{\\bf":[133],"n}:n\\ge":[134],"k\\}$":[135],"family":[138],"by":[142],"$n$":[143],"\"grids\".":[144]},"counts_by_year":[],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}