{"id":"https://openalex.org/W3088390675","doi":"https://doi.org/10.1137/20m1370112","title":"Parity-Unimodality and a Cyclic Sieving Phenomenon for Necklaces","display_name":"Parity-Unimodality and a Cyclic Sieving Phenomenon for Necklaces","publication_year":2021,"publication_date":"2021-01-01","ids":{"openalex":"https://openalex.org/W3088390675","doi":"https://doi.org/10.1137/20m1370112","mag":"3088390675"},"language":"en","primary_location":{"id":"doi:10.1137/20m1370112","is_oa":false,"landing_page_url":"https://doi.org/10.1137/20m1370112","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1812.04578","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5054558507","display_name":"Eric Nathan Stucky","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Eric N. Stucky","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5054558507"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.10271903,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"35","issue":"3","first_page":"2049","last_page":"2069"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":1.0,"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":1.0,"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/T10287","display_name":"Algebraic structures and combinatorial models","score":0.9980999827384949,"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.9973999857902527,"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/unimodality","display_name":"Unimodality","score":0.8154306411743164},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7223024964332581},{"id":"https://openalex.org/keywords/phenomenon","display_name":"Phenomenon","score":0.7004489898681641},{"id":"https://openalex.org/keywords/parity","display_name":"Parity (physics)","score":0.6670567989349365},{"id":"https://openalex.org/keywords/catalan-number","display_name":"Catalan number","score":0.6572651863098145},{"id":"https://openalex.org/keywords/transitive-relation","display_name":"Transitive relation","score":0.6474561095237732},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6298637390136719},{"id":"https://openalex.org/keywords/macdonald-polynomials","display_name":"Macdonald polynomials","score":0.5338160991668701},{"id":"https://openalex.org/keywords/irrational-number","display_name":"Irrational number","score":0.5094242691993713},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.506065309047699},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.4348228871822357},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3481348752975464},{"id":"https://openalex.org/keywords/discrete-orthogonal-polynomials","display_name":"Discrete orthogonal polynomials","score":0.3043648600578308},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.10762098431587219},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.04837316274642944}],"concepts":[{"id":"https://openalex.org/C21596040","wikidata":"https://www.wikidata.org/wiki/Q2896771","display_name":"Unimodality","level":2,"score":0.8154306411743164},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7223024964332581},{"id":"https://openalex.org/C50335755","wikidata":"https://www.wikidata.org/wiki/Q483247","display_name":"Phenomenon","level":2,"score":0.7004489898681641},{"id":"https://openalex.org/C2777151079","wikidata":"https://www.wikidata.org/wiki/Q141160","display_name":"Parity (physics)","level":2,"score":0.6670567989349365},{"id":"https://openalex.org/C43929256","wikidata":"https://www.wikidata.org/wiki/Q270513","display_name":"Catalan number","level":2,"score":0.6572651863098145},{"id":"https://openalex.org/C191399111","wikidata":"https://www.wikidata.org/wiki/Q64861","display_name":"Transitive relation","level":2,"score":0.6474561095237732},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6298637390136719},{"id":"https://openalex.org/C171686973","wikidata":"https://www.wikidata.org/wiki/Q17098676","display_name":"Macdonald polynomials","level":4,"score":0.5338160991668701},{"id":"https://openalex.org/C94931360","wikidata":"https://www.wikidata.org/wiki/Q607728","display_name":"Irrational number","level":2,"score":0.5094242691993713},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.506065309047699},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.4348228871822357},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3481348752975464},{"id":"https://openalex.org/C52537462","wikidata":"https://www.wikidata.org/wiki/Q5282055","display_name":"Discrete orthogonal polynomials","level":3,"score":0.3043648600578308},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.10762098431587219},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.04837316274642944},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C109214941","wikidata":"https://www.wikidata.org/wiki/Q18334","display_name":"Particle physics","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/20m1370112","is_oa":false,"landing_page_url":"https://doi.org/10.1137/20m1370112","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1812.04578","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1812.04578","pdf_url":"https://arxiv.org/pdf/1812.04578","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"mag:3088390675","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1812.04578v3","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1812.04578","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1812.04578","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1812.04578","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1812.04578","pdf_url":"https://arxiv.org/pdf/1812.04578","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1933914410","display_name":null,"funder_award_id":"DMS-1601961","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/W3088390675.pdf","grobid_xml":"https://content.openalex.org/works/W3088390675.grobid-xml"},"referenced_works_count":25,"referenced_works":["https://openalex.org/W1533562873","https://openalex.org/W1634042400","https://openalex.org/W1637800604","https://openalex.org/W1776330888","https://openalex.org/W1849976071","https://openalex.org/W1988847596","https://openalex.org/W2001482097","https://openalex.org/W2001613603","https://openalex.org/W2004552730","https://openalex.org/W2013327405","https://openalex.org/W2054210513","https://openalex.org/W2057990722","https://openalex.org/W2068045918","https://openalex.org/W2068532552","https://openalex.org/W2148832297","https://openalex.org/W2268812832","https://openalex.org/W2516083661","https://openalex.org/W2791702098","https://openalex.org/W2798968480","https://openalex.org/W2962851172","https://openalex.org/W2963437753","https://openalex.org/W2963888294","https://openalex.org/W2999197910","https://openalex.org/W3105001877","https://openalex.org/W3105277314"],"related_works":["https://openalex.org/W3200482042","https://openalex.org/W3004374249","https://openalex.org/W1451989222","https://openalex.org/W2551404887","https://openalex.org/W2543754176","https://openalex.org/W3134269267","https://openalex.org/W1975069680","https://openalex.org/W3099938692","https://openalex.org/W2021530972","https://openalex.org/W2397859226","https://openalex.org/W2952193320","https://openalex.org/W3013525889","https://openalex.org/W2769849871","https://openalex.org/W3000541016","https://openalex.org/W2517598312","https://openalex.org/W2158636704","https://openalex.org/W3047994296","https://openalex.org/W3040488170","https://openalex.org/W2029399622","https://openalex.org/W2963656636"],"abstract_inverted_index":{"We":[0],"discuss":[1],"two":[2],"surprising":[3],"properties":[4],"of":[5,8,61],"a":[6,53,58,62],"family":[7],"polynomials":[9,33],"that":[10,29,38,50],"generalize":[11],"the":[12,19,30,39],"Mahonian":[13],"$q$-Catalan":[14],"polynomials,":[15],"and":[16,41],"more":[17,63],"generally":[18],"$q$-Schr\u00f6der":[20,32],"polynomials.":[21],"By":[22],"interpreting":[23],"them":[24],"as":[25],"$\\mathfrak{sl}_2$-characters,":[26],"we":[27,48],"show":[28,49],"rational":[31],"are":[34,44],"parity-unimodal,":[35],"which":[36],"means":[37],"even-":[40],"odd-degree":[42],"coefficients":[43],"separately":[45],"unimodal.":[46],"Second,":[47],"they":[51],"exhibit":[52],"$q=-1$":[54],"phenomenon.":[55],"This":[56],"is":[57],"special":[59],"case":[60],"general":[64],"cyclic":[65],"sieving":[66],"phenomenon":[67],"for":[68],"certain":[69],"transitive":[70],"$S_n$-actions,":[71],"deduced":[72],"from":[73],"Molien's":[74],"formula.":[75]},"counts_by_year":[],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}