{"id":"https://openalex.org/W2357603150","doi":"https://doi.org/10.37236/5950","title":"Unimodality via Alternating Gamma Vectors","display_name":"Unimodality via Alternating Gamma Vectors","publication_year":2016,"publication_date":"2016-05-27","ids":{"openalex":"https://openalex.org/W2357603150","doi":"https://doi.org/10.37236/5950","mag":"2357603150"},"language":"en","primary_location":{"id":"doi:10.37236/5950","is_oa":true,"landing_page_url":"https://doi.org/10.37236/5950","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v23i2p40/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/v23i2p40/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5081193193","display_name":"Charles Brittenham","orcid":null},"institutions":[{"id":"https://openalex.org/I118353179","display_name":"DePaul University","ror":"https://ror.org/04xtx5t16","country_code":"US","type":"education","lineage":["https://openalex.org/I118353179"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Charles Brittenham","raw_affiliation_strings":["DePaul University"],"affiliations":[{"raw_affiliation_string":"DePaul University","institution_ids":["https://openalex.org/I118353179"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5030067801","display_name":"Andrew T. Carroll","orcid":null},"institutions":[{"id":"https://openalex.org/I118353179","display_name":"DePaul University","ror":"https://ror.org/04xtx5t16","country_code":"US","type":"education","lineage":["https://openalex.org/I118353179"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Andrew T. Carroll","raw_affiliation_strings":["DePaul University"],"affiliations":[{"raw_affiliation_string":"DePaul University","institution_ids":["https://openalex.org/I118353179"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5015920373","display_name":"T. Kyle Petersen","orcid":"https://orcid.org/0000-0001-5188-6049"},"institutions":[{"id":"https://openalex.org/I118353179","display_name":"DePaul University","ror":"https://ror.org/04xtx5t16","country_code":"US","type":"education","lineage":["https://openalex.org/I118353179"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"T. Kyle Petersen","raw_affiliation_strings":["DePaul University"],"affiliations":[{"raw_affiliation_string":"DePaul University","institution_ids":["https://openalex.org/I118353179"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5061284665","display_name":"Connor Thomas","orcid":null},"institutions":[{"id":"https://openalex.org/I118353179","display_name":"DePaul University","ror":"https://ror.org/04xtx5t16","country_code":"US","type":"education","lineage":["https://openalex.org/I118353179"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Connor Thomas","raw_affiliation_strings":["DePaul University"],"affiliations":[{"raw_affiliation_string":"DePaul University","institution_ids":["https://openalex.org/I118353179"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5081193193"],"corresponding_institution_ids":["https://openalex.org/I118353179"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.08010226,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"23","issue":"2","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":0.9979000091552734,"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":0.9979000091552734,"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/T11797","display_name":"graph theory and CDMA systems","score":0.9957000017166138,"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"}},{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.992900013923645,"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/unimodality","display_name":"Unimodality","score":0.9773362874984741},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8186120390892029},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7598061561584473},{"id":"https://openalex.org/keywords/binomial-coefficient","display_name":"Binomial coefficient","score":0.6321042776107788},{"id":"https://openalex.org/keywords/sign","display_name":"Sign (mathematics)","score":0.5670075416564941},{"id":"https://openalex.org/keywords/gaussian-binomial-coefficient","display_name":"Gaussian binomial coefficient","score":0.5002048015594482},{"id":"https://openalex.org/keywords/combinatorial-proof","display_name":"Combinatorial proof","score":0.4895380437374115},{"id":"https://openalex.org/keywords/unit-vector","display_name":"Unit vector","score":0.43037787079811096},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.39250463247299194},{"id":"https://openalex.org/keywords/negative-binomial-distribution","display_name":"Negative binomial distribution","score":0.10959729552268982},{"id":"https://openalex.org/keywords/poisson-distribution","display_name":"Poisson distribution","score":0.06719619035720825},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.0665023922920227},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.05636194348335266}],"concepts":[{"id":"https://openalex.org/C21596040","wikidata":"https://www.wikidata.org/wiki/Q2896771","display_name":"Unimodality","level":2,"score":0.9773362874984741},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8186120390892029},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7598061561584473},{"id":"https://openalex.org/C34718186","wikidata":"https://www.wikidata.org/wiki/Q209875","display_name":"Binomial coefficient","level":2,"score":0.6321042776107788},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.5670075416564941},{"id":"https://openalex.org/C139505817","wikidata":"https://www.wikidata.org/wiki/Q5527834","display_name":"Gaussian binomial coefficient","level":4,"score":0.5002048015594482},{"id":"https://openalex.org/C118539577","wikidata":"https://www.wikidata.org/wiki/Q3402784","display_name":"Combinatorial proof","level":2,"score":0.4895380437374115},{"id":"https://openalex.org/C84316537","wikidata":"https://www.wikidata.org/wiki/Q36255","display_name":"Unit vector","level":2,"score":0.43037787079811096},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.39250463247299194},{"id":"https://openalex.org/C199335787","wikidata":"https://www.wikidata.org/wiki/Q743364","display_name":"Negative binomial distribution","level":3,"score":0.10959729552268982},{"id":"https://openalex.org/C100906024","wikidata":"https://www.wikidata.org/wiki/Q205692","display_name":"Poisson distribution","level":2,"score":0.06719619035720825},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0665023922920227},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.05636194348335266}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/5950","is_oa":true,"landing_page_url":"https://doi.org/10.37236/5950","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v23i2p40/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:10.37236/5950","is_oa":true,"landing_page_url":"https://doi.org/10.37236/5950","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v23i2p40/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":[],"funders":[{"id":"https://openalex.org/F4320310248","display_name":"DePaul University","ror":"https://ror.org/04xtx5t16"},{"id":"https://openalex.org/F4320332624","display_name":"College of Science and Health","ror":"https://ror.org/04xtx5t16"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2357603150.pdf","grobid_xml":"https://content.openalex.org/works/W2357603150.grobid-xml"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W123211135","https://openalex.org/W758484163","https://openalex.org/W1485110504","https://openalex.org/W1515761639","https://openalex.org/W1981696777","https://openalex.org/W2041179721","https://openalex.org/W2082899356","https://openalex.org/W2093580742","https://openalex.org/W2094459512","https://openalex.org/W2101865984","https://openalex.org/W2126209209","https://openalex.org/W2167597938","https://openalex.org/W2963155231","https://openalex.org/W4234408173","https://openalex.org/W4238933498","https://openalex.org/W4250083519","https://openalex.org/W6601771515","https://openalex.org/W6602462696","https://openalex.org/W6629251564","https://openalex.org/W6630930139","https://openalex.org/W6637081280","https://openalex.org/W6697549010"],"related_works":["https://openalex.org/W2126042315","https://openalex.org/W2962965200","https://openalex.org/W2123986695","https://openalex.org/W2949232015","https://openalex.org/W4291954501","https://openalex.org/W2071059532","https://openalex.org/W4289376613","https://openalex.org/W1997892198","https://openalex.org/W2952684280","https://openalex.org/W2169367072"],"abstract_inverted_index":{"For":[0],"a":[1,11,21,36,109,119,131,139,149],"polynomial":[2],"with":[3,63],"palindromic":[4],"coefficients,":[5],"unimodality":[6,18],"is":[7,19,71],"equivalent":[8],"to":[9,84,112],"having":[10,20],"nonnegative":[12,22,37],"$g$-vector.":[13,125],"A":[14],"sufficient":[15],"condition":[16],"for":[17,45,68,123,152],"$\\gamma$-vector,":[23],"though":[24],"one":[25],"can":[26,147],"have":[27,35],"negative":[28],"entries":[29,89],"in":[30,52,118],"the":[31,57,80,88,91,100,103,114,124,153,160],"$\\gamma$-vector":[32,83],"and":[33,159],"still":[34],"$g$-vector.In":[38],"this":[39,138],"paper":[40],"we":[41,86,107,129,145],"provide":[42],"combinatorial":[43,65,74,132],"models":[44],"three":[46],"families":[47],"of":[48,76,90,102,105,134,141],"$\\gamma$-vectors":[49,58],"that":[50,144],"alternate":[51],"sign.":[53],"In":[54,99,126],"each":[55],"case,":[56],"come":[59],"from":[60,82],"unimodal":[61],"polynomials":[62],"straightforward":[64,73],"descriptions,":[66],"but":[67,94],"which":[69],"there":[70],"no":[72],"proof":[75,133],"unimodality.":[77,135],"By":[78],"using":[79],"transformation":[81],"$g$-vector,":[85],"express":[87],"$g$-vector":[92],"combinatorially,":[93],"as":[95],"an":[96],"alternating":[97,115],"sum.":[98],"case":[101],"$q$-analogue":[104],"$n!$,":[106],"use":[108],"sign-reversing":[110],"involution":[111],"interpret":[113],"sum,":[116],"resulting":[117],"manifestly":[120],"positive":[121],"formula":[122],"other":[127,154],"words,":[128],"give":[130],"We":[136],"consider":[137],"\"proof":[140],"concept\"":[142],"result":[143,151],"hope":[146],"inspire":[148],"similar":[150],"two":[155],"cases,":[156],"$\\prod_{j=1}^n":[157],"(1+q^j)$":[158],"$q$-binomial":[161],"coefficient":[162],"${n\\brack":[163],"k}$.":[164]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
