{"id":"https://openalex.org/W4282974009","doi":"https://doi.org/10.48550/arxiv.2206.06472","title":"Trimer covers in the triangular grid: twenty mostly open problems","display_name":"Trimer covers in the triangular grid: twenty mostly open problems","publication_year":2022,"publication_date":"2022-06-13","ids":{"openalex":"https://openalex.org/W4282974009","doi":"https://doi.org/10.48550/arxiv.2206.06472"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2206.06472","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2206.06472","pdf_url":"https://arxiv.org/pdf/2206.06472","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2206.06472","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5032538374","display_name":"James Gary Propp","orcid":"https://orcid.org/0000-0002-0913-3649","observed_orcids":["https://orcid.org/0000-0002-0913-3649"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Propp, James","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5032538374"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"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":0.9980000257492065,"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":0.9980000257492065,"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/T11428","display_name":"Advanced Mathematical Identities","score":0.00019999999494757503,"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"}},{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":9.999999747378752e-05,"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/triangular-grids","display_name":"triangular grids","score":0.906000018119812},{"id":"https://openalex.org/keywords/rhombus-tilings","display_name":"rhombus tilings","score":0.7900000214576721},{"id":"https://openalex.org/keywords/domino-tilings","display_name":"domino tilings","score":0.9990000128746033},{"id":"https://openalex.org/keywords/tiling-enumeration","display_name":"tiling enumeration","score":0.6499999761581421},{"id":"https://openalex.org/keywords/aztec-diamond","display_name":"Aztec diamond","score":0.7509999871253967},{"id":"https://openalex.org/keywords/hexagon","display_name":"hexagons","score":0.7590000033378601},{"id":"https://openalex.org/keywords/open-problems","display_name":"open problems","score":0.6690000295639038}],"concepts":[{"id":"https://openalex.org/C161903591","wikidata":"https://www.wikidata.org/wiki/Q41159","display_name":"Rhombus","level":2,"score":0.942224383354187},{"id":"https://openalex.org/C2776403692","wikidata":"https://www.wikidata.org/wiki/Q898416","display_name":"Trimer","level":3,"score":0.8352285623550415},{"id":"https://openalex.org/C21550011","wikidata":"https://www.wikidata.org/wiki/Q7632143","display_name":"Substitution tiling","level":2,"score":0.7874022722244263},{"id":"https://openalex.org/C2776416436","wikidata":"https://www.wikidata.org/wiki/Q3751781","display_name":"Domino","level":3,"score":0.7108990550041199},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6784538626670837},{"id":"https://openalex.org/C35908053","wikidata":"https://www.wikidata.org/wiki/Q3063631","display_name":"Hexagonal tiling","level":3,"score":0.6485665440559387},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.5498098731040955},{"id":"https://openalex.org/C187691185","wikidata":"https://www.wikidata.org/wiki/Q2020720","display_name":"Grid","level":2,"score":0.4639759659767151},{"id":"https://openalex.org/C2776745293","wikidata":"https://www.wikidata.org/wiki/Q7798302","display_name":"Thriving","level":2,"score":0.458524614572525},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4498820900917053},{"id":"https://openalex.org/C2779546866","wikidata":"https://www.wikidata.org/wiki/Q531306","display_name":"Dimer","level":2,"score":0.3913050889968872},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.25403445959091187},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1859196126461029},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.08053842186927795},{"id":"https://openalex.org/C46141821","wikidata":"https://www.wikidata.org/wiki/Q209402","display_name":"Nuclear magnetic resonance","level":1,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C36289849","wikidata":"https://www.wikidata.org/wiki/Q34749","display_name":"Social science","level":1,"score":0.0},{"id":"https://openalex.org/C144024400","wikidata":"https://www.wikidata.org/wiki/Q21201","display_name":"Sociology","level":0,"score":0.0},{"id":"https://openalex.org/C161790260","wikidata":"https://www.wikidata.org/wiki/Q82264","display_name":"Catalysis","level":2,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2206.06472","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2206.06472","pdf_url":"https://arxiv.org/pdf/2206.06472","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2206.06472","doi":"https://doi.org/10.48550/arxiv.2206.06472","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2206.06472","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2206.06472","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2206.06472","pdf_url":"https://arxiv.org/pdf/2206.06472","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4282974009.pdf","grobid_xml":"https://content.openalex.org/works/W4282974009.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W3206739003","https://openalex.org/W4286901852","https://openalex.org/W2952996678","https://openalex.org/W2093102771","https://openalex.org/W4282974009","https://openalex.org/W2950147731","https://openalex.org/W2179629055","https://openalex.org/W2739542823","https://openalex.org/W2254091502","https://openalex.org/W1985925313"],"abstract_inverted_index":{"In":[0,38],"the":[1,5,30,82],"past":[2],"three":[3],"decades,":[4],"study":[6,31],"of":[7,13,22,32,35,84],"rhombus":[8,62],"tilings":[9,12,63],"and":[10,64],"domino":[11,68],"various":[14],"plane":[15,50],"regions":[16,51],"has":[17],"been":[18],"a":[19,56],"thriving":[20],"subfield":[21],"enumerative":[23],"combinatorics.":[24],"Physicists":[25],"classify":[26],"such":[27],"work":[28],"as":[29,71],"dimer":[33,44],"covers":[34,45],"finite":[36],"graphs.":[37],"this":[39],"article":[40],"we":[41],"move":[42],"beyond":[43],"to":[46,59],"trimer":[47],"covers,":[48],"introducing":[49],"called":[52],"benzels":[53],"that":[54],"play":[55],"role":[57],"analogous":[58],"hexagons":[60],"for":[61,67],"Aztec":[65],"diamonds":[66],"tilings,":[69],"inasmuch":[70],"one":[72],"finds":[73],"many":[74],"(so":[75],"far":[76],"mostly":[77],"conjectural)":[78],"exact":[79],"formulas":[80],"governing":[81],"number":[83],"tilings.":[85]},"counts_by_year":[{"year":2023,"cited_by_count":1}],"updated_date":"2026-09-22T06:50:58.402377","created_date":"2025-10-10T00:00:00"}
