{"id":"https://openalex.org/W1842804924","doi":"https://doi.org/10.37236/1841","title":"Enumerative Formulae for Unrooted Planar Maps: a Pattern","display_name":"Enumerative Formulae for Unrooted Planar Maps: a Pattern","publication_year":2004,"publication_date":"2004-12-07","ids":{"openalex":"https://openalex.org/W1842804924","doi":"https://doi.org/10.37236/1841","mag":"1842804924"},"language":"en","primary_location":{"id":"doi:10.37236/1841","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1841","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i1r88/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/v11i1r88/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5073772029","display_name":"Valery A. Liskovets","orcid":"https://orcid.org/0000-0001-5168-1828"},"institutions":[{"id":"https://openalex.org/I4210120266","display_name":"Czech Academy of Sciences, Institute of Mathematics","ror":"https://ror.org/02tv1yf50","country_code":"CZ","type":"facility","lineage":["https://openalex.org/I202391551","https://openalex.org/I4210120266"]}],"countries":["CZ"],"is_corresponding":true,"raw_author_name":"Valery A. Liskovets","raw_affiliation_strings":["Institute of Mathematics National Academy of Sciences, 220072, Minsk, BELARUS"],"affiliations":[{"raw_affiliation_string":"Institute of Mathematics National Academy of Sciences, 220072, Minsk, BELARUS","institution_ids":["https://openalex.org/I4210120266"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5073772029"],"corresponding_institution_ids":["https://openalex.org/I4210120266"],"apc_list":null,"apc_paid":null,"fwci":1.5511,"has_fulltext":true,"cited_by_count":7,"citation_normalized_percentile":{"value":0.78429472,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"11","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":0.9994999766349792,"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":0.9994999766349792,"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/T10304","display_name":"Geometric and Algebraic Topology","score":0.9876000285148621,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T11673","display_name":"Advanced Topics in Algebra","score":0.98089998960495,"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/mathematics","display_name":"Mathematics","score":0.9113708734512329},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7738889455795288},{"id":"https://openalex.org/keywords/automorphism","display_name":"Automorphism","score":0.6146822571754456},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.596426248550415},{"id":"https://openalex.org/keywords/isomorphism","display_name":"Isomorphism (crystallography)","score":0.5693960189819336},{"id":"https://openalex.org/keywords/separable-space","display_name":"Separable space","score":0.5473102331161499},{"id":"https://openalex.org/keywords/generating-function","display_name":"Generating function","score":0.5400342345237732},{"id":"https://openalex.org/keywords/enumeration","display_name":"Enumeration","score":0.49732354283332825},{"id":"https://openalex.org/keywords/eulerian-path","display_name":"Eulerian path","score":0.4934522211551666},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.46451014280319214},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4486178159713745},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.44255346059799194},{"id":"https://openalex.org/keywords/constant-function","display_name":"Constant function","score":0.4378650486469269},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3264678716659546},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.18310540914535522},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.13695326447486877},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.0998135507106781}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9113708734512329},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7738889455795288},{"id":"https://openalex.org/C118712358","wikidata":"https://www.wikidata.org/wiki/Q782566","display_name":"Automorphism","level":2,"score":0.6146822571754456},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.596426248550415},{"id":"https://openalex.org/C203436722","wikidata":"https://www.wikidata.org/wiki/Q902950","display_name":"Isomorphism (crystallography)","level":3,"score":0.5693960189819336},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.5473102331161499},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.5400342345237732},{"id":"https://openalex.org/C156340839","wikidata":"https://www.wikidata.org/wiki/Q2704791","display_name":"Enumeration","level":2,"score":0.49732354283332825},{"id":"https://openalex.org/C43058520","wikidata":"https://www.wikidata.org/wiki/Q624580","display_name":"Eulerian path","level":3,"score":0.4934522211551666},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.46451014280319214},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4486178159713745},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.44255346059799194},{"id":"https://openalex.org/C153658351","wikidata":"https://www.wikidata.org/wiki/Q746264","display_name":"Constant function","level":3,"score":0.4378650486469269},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3264678716659546},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.18310540914535522},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.13695326447486877},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.0998135507106781},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C8010536","wikidata":"https://www.wikidata.org/wiki/Q160398","display_name":"Crystallography","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C115624301","wikidata":"https://www.wikidata.org/wiki/Q895901","display_name":"Crystal structure","level":2,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C53469067","wikidata":"https://www.wikidata.org/wiki/Q505735","display_name":"Lagrangian","level":2,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.37236/1841","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1841","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i1r88/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:10.1.1.139.5645","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.139.5645","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_11/PostScriptfiles/v11i1r88.ps","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/1841","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1841","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i1r88/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/W1842804924.pdf","grobid_xml":"https://content.openalex.org/works/W1842804924.grobid-xml"},"referenced_works_count":29,"referenced_works":["https://openalex.org/W199076941","https://openalex.org/W1972550776","https://openalex.org/W1975206347","https://openalex.org/W1979595999","https://openalex.org/W1986886622","https://openalex.org/W1995036450","https://openalex.org/W1996008071","https://openalex.org/W2002806015","https://openalex.org/W2009138921","https://openalex.org/W2021355739","https://openalex.org/W2022470848","https://openalex.org/W2032674496","https://openalex.org/W2040783325","https://openalex.org/W2042456974","https://openalex.org/W2047466759","https://openalex.org/W2056893013","https://openalex.org/W2066480898","https://openalex.org/W2076224248","https://openalex.org/W2088683615","https://openalex.org/W2102928439","https://openalex.org/W2149293266","https://openalex.org/W2164323648","https://openalex.org/W2286443079","https://openalex.org/W2313365477","https://openalex.org/W2331690363","https://openalex.org/W2335491319","https://openalex.org/W2908167248","https://openalex.org/W4210345609","https://openalex.org/W6608094227"],"related_works":["https://openalex.org/W1982980084","https://openalex.org/W2024532720","https://openalex.org/W2994629112","https://openalex.org/W4297416234","https://openalex.org/W1996930411","https://openalex.org/W2953640051","https://openalex.org/W1676059255","https://openalex.org/W2971379563","https://openalex.org/W2950351494","https://openalex.org/W1753623017"],"abstract_inverted_index":{"We":[0],"present":[1],"uniformly":[2],"available":[3],"simple":[4,116],"enumerative":[5,118],"formulae":[6,31],"for":[7,122],"unrooted":[8,138],"planar":[9,139],"$n$-edge":[10],"maps":[11,25],"(counted":[12],"up":[13],"to":[14,33,39,55,62,102,130],"orientation-preserving":[15],"isomorphism)":[16],"of":[17,42,58,64,113,137],"numerous":[18],"classes":[19],"including":[20],"arbitrary,":[21],"loopless,":[22],"non-separable,":[23],"eulerian":[24,124],"and":[26,78,83,134],"plane":[27],"trees.":[28],"All":[29],"the":[30,40,43,59,65,74,95,97],"conform":[32],"a":[34,86,110],"certain":[35],"pattern":[36],"with":[37],"respect":[38],"terms":[41],"sum":[44],"over":[45],"$t\\mid":[46],"n,\\,t\\!":[47],"&lt;":[48],"\\!n.$":[49],"Namely,":[50],"these":[51],"terms,":[52],"which":[53,108],"correspond":[54],"non-trivial":[56],"automorphisms":[57],"maps,":[60],"prove":[61],"be":[63],"form":[66],"$\\phi\\left({n\\over":[67],"t}\\right)\\alpha\\,r^t":[68],"{k\\,t\\choose":[69],"t}$,":[70],"where":[71],"$\\phi(m)$":[72],"is":[73,85,109,129],"Euler":[75],"function,":[76],"$k$":[77],"$r$":[79],"are":[80,120],"integer":[81],"constants":[82],"$\\alpha$":[84],"constant":[87],"or":[88],"takes":[89],"only":[90],"two":[91],"rational":[92,111],"values.":[93],"On":[94],"contrary,":[96],"main,":[98],"\"rooted\"":[99],"summand":[100],"corresponding":[101],"$t=n$":[103],"contains":[104],"an":[105],"additional":[106],"factor":[107],"function":[112],"$n$.":[114],"Two":[115],"new":[117],"results":[119,136],"deduced":[121],"bicolored":[123],"maps.":[125],"A":[126],"collateral":[127],"aim":[128],"briefly":[131],"survey":[132],"recent":[133],"old":[135],"map":[140],"enumeration.":[141]},"counts_by_year":[{"year":2017,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
