{"id":"https://openalex.org/W1520897746","doi":"https://doi.org/10.37236/1093","title":"Hard Squares with Negative Activity and Rhombus Tilings of the Plane","display_name":"Hard Squares with Negative Activity and Rhombus Tilings of the Plane","publication_year":2006,"publication_date":"2006-08-07","ids":{"openalex":"https://openalex.org/W1520897746","doi":"https://doi.org/10.37236/1093","mag":"1520897746"},"language":"en","primary_location":{"id":"doi:10.37236/1093","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1093","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v13i1r67/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/v13i1r67/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5111492318","display_name":"Jakob J\u00f6nsson","orcid":null},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Jakob Jonsson","raw_affiliation_strings":["Department of Mathematics Massachusetts Institute of Technology, Cambridge, MA 02139"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics Massachusetts Institute of Technology, Cambridge, MA 02139","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5111492318"],"corresponding_institution_ids":["https://openalex.org/I63966007"],"apc_list":null,"apc_paid":null,"fwci":5.3923,"has_fulltext":true,"cited_by_count":42,"citation_normalized_percentile":{"value":0.95927482,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"13","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9998000264167786,"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/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9998000264167786,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9986000061035156,"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9970999956130981,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.808030366897583},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7858583927154541},{"id":"https://openalex.org/keywords/rhombus","display_name":"Rhombus","score":0.7744271755218506},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.621694803237915},{"id":"https://openalex.org/keywords/coprime-integers","display_name":"Coprime integers","score":0.6001441478729248},{"id":"https://openalex.org/keywords/modulo","display_name":"Modulo","score":0.46409955620765686},{"id":"https://openalex.org/keywords/eulers-formula","display_name":"Euler's formula","score":0.4507763087749481},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.2692270874977112},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2129080593585968}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.808030366897583},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7858583927154541},{"id":"https://openalex.org/C161903591","wikidata":"https://www.wikidata.org/wiki/Q41159","display_name":"Rhombus","level":2,"score":0.7744271755218506},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.621694803237915},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.6001441478729248},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.46409955620765686},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.4507763087749481},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2692270874977112},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2129080593585968}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.37236/1093","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1093","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v13i1r67/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:10.1.1.92.2977","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.92.2977","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.emis.de/journals/EJC/Volume_13/PDF/v13i1r67.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/1093","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1093","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v13i1r67/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":[{"id":"https://openalex.org/G7025150052","display_name":null,"funder_award_id":"GRK 588/2","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320324094","display_name":"Technische Universit\u00e4t Berlin","ror":"https://ror.org/03v4gjf40"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1520897746.pdf","grobid_xml":"https://content.openalex.org/works/W1520897746.grobid-xml"},"referenced_works_count":10,"referenced_works":["https://openalex.org/W135448871","https://openalex.org/W2064020260","https://openalex.org/W2070673991","https://openalex.org/W2089209018","https://openalex.org/W2091521721","https://openalex.org/W2115103988","https://openalex.org/W2169447603","https://openalex.org/W3099152892","https://openalex.org/W6605030782","https://openalex.org/W6772086821"],"related_works":["https://openalex.org/W3164467166","https://openalex.org/W1970378093","https://openalex.org/W2584375338","https://openalex.org/W1552842083","https://openalex.org/W2951995717","https://openalex.org/W2371749308","https://openalex.org/W3025068760","https://openalex.org/W4363677397","https://openalex.org/W3164481304","https://openalex.org/W3180944311"],"abstract_inverted_index":{"Let":[0],"$S_{m,n}$":[1],"be":[2],"the":[3,6,46,50,64,98,109,150,155,164,168],"graph":[4],"on":[5],"vertex":[7],"set":[8],"${\\Bbb":[9],"Z}_m":[10],"\\times":[11],"{\\Bbb":[12],"Z}_n$":[13],"in":[14,57,81],"which":[15,123],"there":[16],"is":[17,137],"an":[18],"edge":[19],"between":[20],"$(a,b)$":[21],"and":[22,25,73,88,94],"$(c,d)$":[23],"if":[24,27],"only":[26],"either":[28],"$(a,b)":[29,34],"=":[30,35],"(c,d\\pm":[31],"1)$":[32],"or":[33],"(c":[36],"\\pm":[37],"1,d)$":[38],"modulo":[39],"$(m,n)$.":[40],"We":[41],"present":[42],"a":[43,79,138],"formula":[44],"for":[45],"Euler":[47,66,99,157],"characteristic":[48,67,100,158],"of":[49,54,68,101,108,128,133,152,159,167],"simplicial":[51],"complex":[52],"$\\Sigma_{m,n}$":[53,69,102,160],"independent":[55],"sets":[56],"$S_{m,n}$.":[58],"In":[59,149],"particular,":[60],"we":[61,96,114],"show":[62],"that":[63,125],"unreduced":[65],"vanishes":[70],"whenever":[71],"$m$":[72,93],"$n$":[74],"are":[75,131],"coprime,":[76],"thereby":[77],"settling":[78],"conjecture":[80,117],"statistical":[82,153],"mechanics":[83],"due":[84,118],"to":[85,103,119,143],"Fendley,":[86],"Schoutens":[87],"van":[89],"Eerten.":[90],"For":[91],"general":[92],"$n$,":[95],"relate":[97],"certain":[104,139],"periodic":[105],"rhombus":[106],"tilings":[107],"plane.":[110],"Using":[111],"this":[112],"correspondence,":[113],"settle":[115],"another":[116],"Fendley":[120],"et":[121],"al.,":[122],"states":[124],"all":[126],"roots":[127,132],"$\\det":[129],"(xI-T_m)$":[130],"unity,":[134],"where":[135],"$T_m$":[136],"transfer":[140],"matrix":[141],"associated":[142],"$\\{\\Sigma_{m,n}":[144],":":[145],"n":[146],"\\ge":[147],"1\\}$.":[148],"language":[151],"mechanics,":[154],"reduced":[156],"coincides":[161],"with":[162,173],"minus":[163],"partition":[165],"function":[166],"corresponding":[169],"hard":[170],"square":[171],"model":[172],"activity":[174],"$-1$.":[175]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2017,"cited_by_count":3},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":5}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}