{"id":"https://openalex.org/W2963580840","doi":"https://doi.org/10.37236/101","title":"Littlewood&amp;ndash;Richardson Coefficients and Integrable Tilings","display_name":"Littlewood&amp;ndash;Richardson Coefficients and Integrable Tilings","publication_year":2009,"publication_date":"2009-01-23","ids":{"openalex":"https://openalex.org/W2963580840","doi":"https://doi.org/10.37236/101","mag":"2963580840"},"language":"en","primary_location":{"id":"doi:10.37236/101","is_oa":true,"landing_page_url":"https://doi.org/10.37236/101","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r12/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":null,"license_id":null,"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/v16i1r12/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5006276965","display_name":"Paul Zinn-Justin","orcid":"https://orcid.org/0000-0002-6201-9562"},"institutions":[{"id":"https://openalex.org/I4210119869","display_name":"Laboratoire de Physique Th\u00e9orique et Hautes Energies","ror":"https://ror.org/02mph9k76","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I39804081","https://openalex.org/I4210098836","https://openalex.org/I4210119869"]},{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"funder","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I102197404","display_name":"Universit\u00e9 Paris-Sud","ror":"https://ror.org/028rypz17","country_code":"FR","type":"education","lineage":["https://openalex.org/I102197404"]},{"id":"https://openalex.org/I39804081","display_name":"Sorbonne Universit\u00e9","ror":"https://ror.org/02en5vm52","country_code":"FR","type":"education","lineage":["https://openalex.org/I39804081"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Paul Zinn-Justin","raw_affiliation_strings":["(CNRS, UMR 8626), Univ Paris-Sud, 91405 Orsay Cedex, France; and","LPTHE (CNRS, UMR 7589), Univ Pierre et Marie Curie-Paris6, 75252 Paris Cedex, France,"],"affiliations":[{"raw_affiliation_string":"(CNRS, UMR 8626), Univ Paris-Sud, 91405 Orsay Cedex, France; and","institution_ids":["https://openalex.org/I102197404","https://openalex.org/I1294671590"]},{"raw_affiliation_string":"LPTHE (CNRS, UMR 7589), Univ Pierre et Marie Curie-Paris6, 75252 Paris Cedex, France,","institution_ids":["https://openalex.org/I39804081","https://openalex.org/I4210119869","https://openalex.org/I1294671590"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5006276965"],"corresponding_institution_ids":["https://openalex.org/I102197404","https://openalex.org/I1294671590","https://openalex.org/I39804081","https://openalex.org/I4210119869"],"apc_list":null,"apc_paid":null,"fwci":2.8406,"has_fulltext":true,"cited_by_count":34,"citation_normalized_percentile":{"value":0.90774015,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"16","issue":"1","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":0.9998999834060669,"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":0.9998999834060669,"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":0.9991000294685364,"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/T12748","display_name":"Molecular spectroscopy and chirality","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/1607","display_name":"Spectroscopy"},"field":{"id":"https://openalex.org/fields/16","display_name":"Chemistry"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7614600658416748},{"id":"https://openalex.org/keywords/integrable-system","display_name":"Integrable system","score":0.6899304389953613},{"id":"https://openalex.org/keywords/schur-product-theorem","display_name":"Schur product theorem","score":0.5731066465377808},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.5485719442367554},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5421382784843445},{"id":"https://openalex.org/keywords/factorial","display_name":"Factorial","score":0.5399795174598694},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.5163706541061401},{"id":"https://openalex.org/keywords/coproduct","display_name":"Coproduct","score":0.4823124408721924},{"id":"https://openalex.org/keywords/simplex","display_name":"Simplex","score":0.4335305690765381},{"id":"https://openalex.org/keywords/schur-polynomial","display_name":"Schur polynomial","score":0.41503891348838806},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.401469886302948},{"id":"https://openalex.org/keywords/schur-complement","display_name":"Schur complement","score":0.3327159583568573},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.224444180727005},{"id":"https://openalex.org/keywords/schur-decomposition","display_name":"Schur decomposition","score":0.21365535259246826},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.13152849674224854},{"id":"https://openalex.org/keywords/macdonald-polynomials","display_name":"Macdonald polynomials","score":0.11479675769805908},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08320429921150208},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.06750035285949707}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7614600658416748},{"id":"https://openalex.org/C200741047","wikidata":"https://www.wikidata.org/wiki/Q1957758","display_name":"Integrable system","level":2,"score":0.6899304389953613},{"id":"https://openalex.org/C102591085","wikidata":"https://www.wikidata.org/wiki/Q17103152","display_name":"Schur product theorem","level":5,"score":0.5731066465377808},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.5485719442367554},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5421382784843445},{"id":"https://openalex.org/C183763347","wikidata":"https://www.wikidata.org/wiki/Q120976","display_name":"Factorial","level":2,"score":0.5399795174598694},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.5163706541061401},{"id":"https://openalex.org/C36259021","wikidata":"https://www.wikidata.org/wiki/Q692689","display_name":"Coproduct","level":2,"score":0.4823124408721924},{"id":"https://openalex.org/C62438384","wikidata":"https://www.wikidata.org/wiki/Q331350","display_name":"Simplex","level":2,"score":0.4335305690765381},{"id":"https://openalex.org/C146922319","wikidata":"https://www.wikidata.org/wiki/Q4298935","display_name":"Schur polynomial","level":5,"score":0.41503891348838806},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.401469886302948},{"id":"https://openalex.org/C2731191","wikidata":"https://www.wikidata.org/wiki/Q1778169","display_name":"Schur complement","level":3,"score":0.3327159583568573},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.224444180727005},{"id":"https://openalex.org/C99383703","wikidata":"https://www.wikidata.org/wiki/Q1064218","display_name":"Schur decomposition","level":4,"score":0.21365535259246826},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.13152849674224854},{"id":"https://openalex.org/C171686973","wikidata":"https://www.wikidata.org/wiki/Q17098676","display_name":"Macdonald polynomials","level":4,"score":0.11479675769805908},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08320429921150208},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.06750035285949707},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C86607863","wikidata":"https://www.wikidata.org/wiki/Q5275279","display_name":"Difference polynomials","level":3,"score":0.0},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/101","is_oa":true,"landing_page_url":"https://doi.org/10.37236/101","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r12/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":null,"license_id":null,"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:10.37236/101","is_oa":true,"landing_page_url":"https://doi.org/10.37236/101","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r12/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":null,"license_id":null,"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/G766923599","display_name":null,"funder_award_id":"ANR-05","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"}],"funders":[{"id":"https://openalex.org/F4320320883","display_name":"Agence Nationale de la Recherche","ror":"https://ror.org/00rbzpz17"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2963580840.pdf","grobid_xml":"https://content.openalex.org/works/W2963580840.grobid-xml"},"referenced_works_count":20,"referenced_works":["https://openalex.org/W1619965796","https://openalex.org/W1807235826","https://openalex.org/W1983326071","https://openalex.org/W1997135482","https://openalex.org/W1998481279","https://openalex.org/W2030304270","https://openalex.org/W2074703608","https://openalex.org/W2091216084","https://openalex.org/W2094594266","https://openalex.org/W2108491571","https://openalex.org/W2124762879","https://openalex.org/W2135893164","https://openalex.org/W2144547618","https://openalex.org/W2155082294","https://openalex.org/W2164915394","https://openalex.org/W2317423124","https://openalex.org/W2963642694","https://openalex.org/W3124471301","https://openalex.org/W6638403129","https://openalex.org/W6639415021"],"related_works":["https://openalex.org/W2146459776","https://openalex.org/W4237788129","https://openalex.org/W2522358176","https://openalex.org/W1975201729","https://openalex.org/W1972018154","https://openalex.org/W2082997104","https://openalex.org/W2362303362","https://openalex.org/W2370298131","https://openalex.org/W1621463705","https://openalex.org/W1605303195"],"abstract_inverted_index":{"We":[0],"provide":[1],"direct":[2],"proofs":[3],"of":[4,39,57],"product":[5,23],"and":[6,42],"coproduct":[7],"formulae":[8,38],"for":[9,29,45],"Schur":[10,31,47],"functions":[11],"where":[12],"the":[13,30,54,58],"coefficients":[14],"(Littlewood\u2013Richardson":[15],"coefficients)":[16],"are":[17],"defined":[18],"as":[19],"counting":[20],"puzzles.":[21],"The":[22,49],"formula":[24],"includes":[25],"a":[26],"second":[27],"alphabet":[28],"functions,":[32],"allowing":[33],"in":[34],"particular":[35],"to":[36],"recover":[37],"[Molev\u2013Sagan":[40],"'99]":[41],"[Knutson\u2013Tao":[43],"'03]":[44],"factorial":[46],"functions.":[48],"method":[50],"is":[51],"based":[52],"on":[53],"quantum":[55],"integrability":[56],"underlying":[59],"tiling":[60],"model.":[61]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":3}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2019-07-30T00:00:00"}
