{"id":"https://openalex.org/W7216501068","doi":"https://doi.org/10.48550/arxiv.2610.00966","title":"Zeros and interlacing for multiset Eulerian-Narayana polynomials","display_name":"Zeros and interlacing for multiset Eulerian-Narayana polynomials","publication_year":2026,"publication_date":"2026-10-01","ids":{"openalex":"https://openalex.org/W7216501068","doi":"https://doi.org/10.48550/arxiv.2610.00966"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.00966","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00966","pdf_url":"https://arxiv.org/pdf/2610.00966","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.00966","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5054926232","display_name":"Philip B. Zhang","orcid":"https://orcid.org/0000-0002-7554-3910","observed_orcids":["https://orcid.org/0000-0002-7554-3910"]},"institutions":[{"id":"https://openalex.org/I15062923","display_name":"Tianjin Normal University","ror":"https://ror.org/05x2td559","country_code":"CN","type":"education","lineage":["https://openalex.org/I15062923"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhang, Philip B.","raw_affiliation_strings":["College of Mathematical Sciences & Institute of Mathematics and Interdisciplinary Sciences , Tianjin Normal University , Tianjin 300387 , P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Mathematical Sciences & Institute of Mathematics and Interdisciplinary Sciences , Tianjin Normal University , Tianjin 300387 , P. R. China","institution_ids":["https://openalex.org/I15062923"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5156113846","display_name":"Tongyuan Zhao","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I204553293","display_name":"China University of Petroleum, Beijing","ror":"https://ror.org/041qf4r12","country_code":"CN","type":"education","lineage":["https://openalex.org/I204553293"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhao, Tongyuan","raw_affiliation_strings":["College of Science , China University of Petroleum (Beijing) , Beijing 102249 , P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Science , China University of Petroleum (Beijing) , Beijing 102249 , P. R. China","institution_ids":["https://openalex.org/I204553293"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"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":0.9994999766349792,"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":0.9994999766349792,"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/T11191","display_name":"Mathematical functions and polynomials","score":0.0005000000237487257,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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":9.999999747378752e-05,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/real-rootedness","display_name":"real-rootedness","score":0.8050000071525574},{"id":"https://openalex.org/keywords/forest-decomposition","display_name":"forest decomposition","score":0.7519999742507935},{"id":"https://openalex.org/keywords/lagrange-inversion","display_name":"Lagrange inversion","score":0.9959999918937683},{"id":"https://openalex.org/keywords/gamma-coefficients","display_name":"gamma coefficient","score":0.6859999895095825},{"id":"https://openalex.org/keywords/totally-nonnegative-matrices","display_name":"totally nonnegative matrices","score":0.8579999804496765},{"id":"https://openalex.org/keywords/narayana-polynomials","display_name":"Narayana polynomials","score":0.9610000252723694},{"id":"https://openalex.org/keywords/eulerian-polynomials","display_name":"Eulerian polynomials","score":0.953000009059906},{"id":"https://openalex.org/keywords/generating-polynomials","display_name":"generating polynomials","score":0.7329999804496765}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8705999851226807},{"id":"https://openalex.org/C2779623528","wikidata":"https://www.wikidata.org/wiki/Q864377","display_name":"Multiset","level":2,"score":0.7775999903678894},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.7498000264167786},{"id":"https://openalex.org/C95439218","wikidata":"https://www.wikidata.org/wiki/Q1666173","display_name":"Interlacing","level":2,"score":0.6852999925613403},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5684999823570251},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.46869999170303345},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4634000062942505},{"id":"https://openalex.org/C108408018","wikidata":"https://www.wikidata.org/wiki/Q1124546","display_name":"Laguerre polynomials","level":2,"score":0.4618000090122223},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.44609999656677246},{"id":"https://openalex.org/C124681953","wikidata":"https://www.wikidata.org/wiki/Q339062","display_name":"Decomposition","level":2,"score":0.4244000017642975},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4237000048160553},{"id":"https://openalex.org/C43058520","wikidata":"https://www.wikidata.org/wiki/Q624580","display_name":"Eulerian path","level":3,"score":0.4146000146865845},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.37130001187324524},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.3700999915599823},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.2937999963760376},{"id":"https://openalex.org/C52537462","wikidata":"https://www.wikidata.org/wiki/Q5282055","display_name":"Discrete orthogonal polynomials","level":3,"score":0.2922999858856201},{"id":"https://openalex.org/C74497983","wikidata":"https://www.wikidata.org/wiki/Q1330788","display_name":"Weierstrass factorization theorem","level":3,"score":0.2809999883174896},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.28029999136924744},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"score":0.271699994802475},{"id":"https://openalex.org/C171686973","wikidata":"https://www.wikidata.org/wiki/Q17098676","display_name":"Macdonald polynomials","level":4,"score":0.26829999685287476},{"id":"https://openalex.org/C5056958","wikidata":"https://www.wikidata.org/wiki/Q6430769","display_name":"Koornwinder polynomials","level":5,"score":0.26570001244544983},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.2619999945163727},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.2615000009536743},{"id":"https://openalex.org/C2987642246","wikidata":"https://www.wikidata.org/wiki/Q5452196","display_name":"First order","level":2,"score":0.2606000006198883},{"id":"https://openalex.org/C2776372474","wikidata":"https://www.wikidata.org/wiki/Q508291","display_name":"Simplicity","level":2,"score":0.257099986076355},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.2533000111579895}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.00966","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00966","pdf_url":"https://arxiv.org/pdf/2610.00966","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.00966","doi":"https://doi.org/10.48550/arxiv.2610.00966","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.00966","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.00966","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00966","pdf_url":"https://arxiv.org/pdf/2610.00966","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G4338915643","display_name":null,"funder_award_id":"12201641","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6545595343","display_name":null,"funder_award_id":"12171362","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7610362479","display_name":null,"funder_award_id":"25JCYBJC00430","funder_id":"https://openalex.org/F4320323993","funder_display_name":"Natural Science Foundation of Tianjin City"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320323993","display_name":"Natural Science Foundation of Tianjin City","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7216501068.pdf","grobid_xml":"https://content.openalex.org/works/W7216501068.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4207002276","https://openalex.org/W3103444687","https://openalex.org/W4403975920","https://openalex.org/W2268659875","https://openalex.org/W4389787543","https://openalex.org/W4407634921","https://openalex.org/W2062912738","https://openalex.org/W2799552353","https://openalex.org/W4416255287","https://openalex.org/W4320733580"],"abstract_inverted_index":{"We":[0,48,82],"prove":[1,84],"the":[2,25,45,55,68,88,98],"real-rootedness":[3],"conjecture":[4],"of":[5,20,44,87,90,97],"Lin,":[6],"Ma,":[7,8],"and":[9,18,32,40,58,70],"Zhou":[10],"(2021)":[11],"for":[12],"multiset":[13],"Eulerian-Narayana":[14],"polynomials,":[15],"including":[16],"simplicity":[17],"negativity":[19],"all":[21],"zeros":[22,89],"after":[23],"removing":[24],"factor":[26],"t.":[27],"Using":[28],"a":[29,37,41],"forest":[30],"decomposition":[31],"Lagrange":[33],"inversion,":[34],"we":[35],"derive":[36],"differential":[38,46],"formula":[39],"first-order":[42],"factorization":[43,75],"operator.":[47],"use":[49],"these":[50],"factors":[51],"to":[52],"show":[53],"that":[54],"nonconstant":[56],"gamma-coefficients":[57],"ordered":[59],"gamma-zeros":[60],"are":[61],"nondecreasing,":[62],"with":[63],"corresponding":[64],"bounds":[65],"given":[66],"by":[67],"Narayana":[69],"Eulerian":[71],"polynomials.":[72],"The":[73],"same":[74],"gives":[76],"totally":[77],"nonnegative":[78],"coefficient":[79],"transition":[80,99],"matrices.":[81,100],"also":[83],"strict":[85],"interlacing":[86],"generating":[91],"polynomials":[92],"arising":[93],"from":[94],"adjacent":[95],"columns":[96]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-10-03T00:00:00"}