{"id":"https://openalex.org/W2962949727","doi":"https://doi.org/10.37236/5770","title":"Mixed Volumes of Hypersimplices","display_name":"Mixed Volumes of Hypersimplices","publication_year":2016,"publication_date":"2016-08-05","ids":{"openalex":"https://openalex.org/W2962949727","doi":"https://doi.org/10.37236/5770","mag":"2962949727"},"language":"en","primary_location":{"id":"doi:10.37236/5770","is_oa":true,"landing_page_url":"https://doi.org/10.37236/5770","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v23i3p19/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/v23i3p19/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5054884992","display_name":"Gaku Liu","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":"Gaku Liu","raw_affiliation_strings":["Massachusetts Institute of Technology"],"affiliations":[{"raw_affiliation_string":"Massachusetts Institute of Technology","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5054884992"],"corresponding_institution_ids":["https://openalex.org/I63966007"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.25140977,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"23","issue":"3","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":0.9972000122070312,"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":0.9972000122070312,"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":0.9962000250816345,"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/T14500","display_name":"Advanced Mathematical Theories","score":0.9312999844551086,"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/eulerian-path","display_name":"Eulerian path","score":0.9246740937232971},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8651585578918457},{"id":"https://openalex.org/keywords/catalan-number","display_name":"Catalan number","score":0.7987717390060425},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6191270351409912},{"id":"https://openalex.org/keywords/interpretation","display_name":"Interpretation (philosophy)","score":0.5501641631126404},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.48335903882980347},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4170071482658386},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.1572684943675995},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.07236549258232117},{"id":"https://openalex.org/keywords/lagrangian","display_name":"Lagrangian","score":0.050865113735198975}],"concepts":[{"id":"https://openalex.org/C43058520","wikidata":"https://www.wikidata.org/wiki/Q624580","display_name":"Eulerian path","level":3,"score":0.9246740937232971},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8651585578918457},{"id":"https://openalex.org/C43929256","wikidata":"https://www.wikidata.org/wiki/Q270513","display_name":"Catalan number","level":2,"score":0.7987717390060425},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6191270351409912},{"id":"https://openalex.org/C527412718","wikidata":"https://www.wikidata.org/wiki/Q855395","display_name":"Interpretation (philosophy)","level":2,"score":0.5501641631126404},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.48335903882980347},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4170071482658386},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.1572684943675995},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.07236549258232117},{"id":"https://openalex.org/C53469067","wikidata":"https://www.wikidata.org/wiki/Q505735","display_name":"Lagrangian","level":2,"score":0.050865113735198975},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/5770","is_oa":true,"landing_page_url":"https://doi.org/10.37236/5770","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v23i3p19/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:10.37236/5770","is_oa":true,"landing_page_url":"https://doi.org/10.37236/5770","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v23i3p19/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":[{"id":"https://openalex.org/G3526151021","display_name":null,"funder_award_id":"1122374","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5635854791","display_name":null,"funder_award_id":"112237","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6894402473","display_name":null,"funder_award_id":"Fellowship","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7297461854","display_name":null,"funder_award_id":"122374","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G848032724","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2962949727.pdf","grobid_xml":"https://content.openalex.org/works/W2962949727.grobid-xml"},"referenced_works_count":6,"referenced_works":["https://openalex.org/W1761786091","https://openalex.org/W2110806960","https://openalex.org/W2130848020","https://openalex.org/W2143023483","https://openalex.org/W3037491991","https://openalex.org/W6630949001"],"related_works":["https://openalex.org/W4295292729","https://openalex.org/W2951263673","https://openalex.org/W1958323954","https://openalex.org/W4300123639","https://openalex.org/W2246177837","https://openalex.org/W3134640962","https://openalex.org/W2962949727","https://openalex.org/W2122442663","https://openalex.org/W2896175204","https://openalex.org/W4289376923"],"abstract_inverted_index":{"In":[0,55],"this":[1],"paper":[2],"we":[3,57,86],"consider":[4,87],"mixed":[5,46,61,79,93],"volumes":[6],"of":[7,9,68,78,92],"combinations":[8],"hypersimplices.":[10],"These":[11],"numbers,":[12,33,35],"called":[13],"\"mixed":[14],"Eulerian":[15,32,47,62,80,94],"numbers\",":[16],"were":[17,24],"first":[18],"considered":[19],"by":[20],"A.":[21],"Postnikov":[22],"and":[23,49,96],"shown":[25],"to":[26,31],"satisfy":[27],"many":[28],"properties":[29,53,77],"related":[30],"Catalan":[34],"binomial":[36],"coefficients,":[37],"etc.":[38],"We":[39,72],"give":[40,97],"a":[41,65,88],"general":[42],"combinatorial":[43,100],"interpretation":[44,101],"for":[45,102],"numbers":[48,81,95],"prove":[50,74],"the":[51],"above":[52],"combinatorially.":[54],"particular,":[56],"show":[58],"that":[59],"each":[60],"number":[63],"enumerates":[64],"certain":[66],"set":[67],"permutations":[69],"in":[70],"$S_n$.":[71],"also":[73],"several":[75],"new":[76],"using":[82],"our":[83],"methods.":[84],"Finally,":[85],"type":[89],"B":[90],"analogue":[91],"an":[98],"analogous":[99],"these":[103],"numbers.":[104]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}