{"id":"https://openalex.org/W1606430465","doi":"https://doi.org/10.37236/1821","title":"Conjectured Combinatorial Models for the Hilbert Series of Generalized Diagonal Harmonics Modules","display_name":"Conjectured Combinatorial Models for the Hilbert Series of Generalized Diagonal Harmonics Modules","publication_year":2004,"publication_date":"2004-09-24","ids":{"openalex":"https://openalex.org/W1606430465","doi":"https://doi.org/10.37236/1821","mag":"1606430465"},"language":"en","primary_location":{"id":"doi:10.37236/1821","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1821","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i1r68/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/v11i1r68/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5081049009","display_name":"Nicholas A. Loehr","orcid":"https://orcid.org/0000-0003-0684-0310"},"institutions":[{"id":"https://openalex.org/I79576946","display_name":"University of Pennsylvania","ror":"https://ror.org/00b30xv10","country_code":"US","type":"education","lineage":["https://openalex.org/I79576946"]},{"id":"https://openalex.org/I36258959","display_name":"University of California, San Diego","ror":"https://ror.org/0168r3w48","country_code":"US","type":"education","lineage":["https://openalex.org/I36258959"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Nicholas A. Loehr","raw_affiliation_strings":["Department of Mathematics University of Pennsylvania Philadelphia, PA 19104","Department of Mathematics University of California at San Diego La Jolla, CA 92093"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics University of Pennsylvania Philadelphia, PA 19104","institution_ids":["https://openalex.org/I79576946"]},{"raw_affiliation_string":"Department of Mathematics University of California at San Diego La Jolla, CA 92093","institution_ids":["https://openalex.org/I36258959"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5113511227","display_name":"Jeffrey B. Remmel","orcid":null},"institutions":[{"id":"https://openalex.org/I36258959","display_name":"University of California, San Diego","ror":"https://ror.org/0168r3w48","country_code":"US","type":"education","lineage":["https://openalex.org/I36258959"]},{"id":"https://openalex.org/I79576946","display_name":"University of Pennsylvania","ror":"https://ror.org/00b30xv10","country_code":"US","type":"education","lineage":["https://openalex.org/I79576946"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jeffrey B. Remmel","raw_affiliation_strings":["Department of Mathematics University of Pennsylvania Philadelphia, PA 19104","Department of Mathematics University of California at San Diego La Jolla, CA 92093"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics University of Pennsylvania Philadelphia, PA 19104","institution_ids":["https://openalex.org/I79576946"]},{"raw_affiliation_string":"Department of Mathematics University of California at San Diego La Jolla, CA 92093","institution_ids":["https://openalex.org/I36258959"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5081049009"],"corresponding_institution_ids":["https://openalex.org/I36258959","https://openalex.org/I79576946"],"apc_list":null,"apc_paid":null,"fwci":3.1108,"has_fulltext":true,"cited_by_count":17,"citation_normalized_percentile":{"value":0.89578284,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":95},"biblio":{"volume":"11","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":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/T11428","display_name":"Advanced Mathematical Identities","score":0.9976000189781189,"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.9883000254631042,"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/mathematics","display_name":"Mathematics","score":0.8552670478820801},{"id":"https://openalex.org/keywords/diagonal","display_name":"Diagonal","score":0.738867461681366},{"id":"https://openalex.org/keywords/bijection","display_name":"Bijection","score":0.7350530624389648},{"id":"https://openalex.org/keywords/bijection-injection-and-surjection","display_name":"Bijection, injection and surjection","score":0.7317584753036499},{"id":"https://openalex.org/keywords/generating-function","display_name":"Generating function","score":0.7130783796310425},{"id":"https://openalex.org/keywords/hilbert\u2013poincar\u00e9-series","display_name":"Hilbert\u2013Poincar\u00e9 series","score":0.5392137765884399},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.5216196775436401},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.5191559195518494},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.5044258832931519},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5035445094108582},{"id":"https://openalex.org/keywords/combinatorial-proof","display_name":"Combinatorial proof","score":0.4370035231113434}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8552670478820801},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.738867461681366},{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.7350530624389648},{"id":"https://openalex.org/C48659774","wikidata":"https://www.wikidata.org/wiki/Q4907197","display_name":"Bijection, injection and surjection","level":3,"score":0.7317584753036499},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.7130783796310425},{"id":"https://openalex.org/C185982889","wikidata":"https://www.wikidata.org/wiki/Q5761255","display_name":"Hilbert\u2013Poincar\u00e9 series","level":2,"score":0.5392137765884399},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.5216196775436401},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5191559195518494},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.5044258832931519},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5035445094108582},{"id":"https://openalex.org/C118539577","wikidata":"https://www.wikidata.org/wiki/Q3402784","display_name":"Combinatorial proof","level":2,"score":0.4370035231113434},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.37236/1821","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1821","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i1r68/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"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.77.9940","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.77.9940","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.combinatorics.org/Volume_11/PDF/v11i1r68.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/1821","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1821","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i1r68/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":[{"display_name":"Sustainable cities and communities","id":"https://metadata.un.org/sdg/11","score":0.5400000214576721}],"awards":[{"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/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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1606430465.pdf","grobid_xml":"https://content.openalex.org/works/W1606430465.grobid-xml"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W43758212","https://openalex.org/W1518468345","https://openalex.org/W1671066123","https://openalex.org/W1970024668","https://openalex.org/W1989735024","https://openalex.org/W2013269097","https://openalex.org/W2022892305","https://openalex.org/W2042843318","https://openalex.org/W2048202079","https://openalex.org/W2050156913","https://openalex.org/W2059036144","https://openalex.org/W2078152973","https://openalex.org/W2087726143","https://openalex.org/W2087941871","https://openalex.org/W2096502393","https://openalex.org/W2128113555","https://openalex.org/W2134984950","https://openalex.org/W2139972197","https://openalex.org/W2140141642","https://openalex.org/W2186344661","https://openalex.org/W3000541016","https://openalex.org/W4239240946","https://openalex.org/W4241337807","https://openalex.org/W6662402918"],"related_works":["https://openalex.org/W2952259524","https://openalex.org/W2160566290","https://openalex.org/W2806276129","https://openalex.org/W4299611798","https://openalex.org/W2963449060","https://openalex.org/W1893226401","https://openalex.org/W3083163437","https://openalex.org/W2042216900","https://openalex.org/W4286421839","https://openalex.org/W4321473432"],"abstract_inverted_index":{"Haglund":[0,78],"and":[1,79],"Loehr":[2],"previously":[3],"conjectured":[4],"two":[5],"equivalent":[6,47],"combinatorial":[7,40,141],"formulas":[8,20,65,142],"for":[9,101,105,143],"the":[10,14,49,55,68,102,113,127,131,144,151],"Hilbert":[11,145],"series":[12,146],"of":[13,24,77,89,112,115,118,147,150],"Garsia-Haiman":[15],"diagonal":[16,152],"harmonics":[17,153],"modules.":[18,154],"These":[19,122],"involve":[21],"weighted":[22],"sums":[23],"labelled":[25,59,90],"Dyck":[26,60],"paths":[27,61,92],"(or":[28],"parking":[29],"functions)":[30],"relative":[31],"to":[32,45,48],"suitable":[33],"statistics.":[34,107],"This":[35],"article":[36],"introduces":[37],"a":[38,98],"third":[39],"formula":[41,100],"that":[42,54,125],"is":[43],"shown":[44],"be":[46],"first":[50],"two.":[51],"We":[52,81,96,108],"show":[53],"four":[56],"statistics":[57,85,129],"on":[58,86],"appearing":[62],"in":[63,94],"these":[64,106,136],"all":[66,126],"have":[67,130],"same":[69,132],"univariate":[70,133],"distribution,":[71],"which":[72],"settles":[73],"an":[74],"earlier":[75],"question":[76],"Loehr.":[80],"then":[82],"introduce":[83],"analogous":[84],"other":[87],"collections":[88],"lattice":[91],"contained":[93],"trapezoids.":[95],"obtain":[97],"fermionic":[99],"generating":[103,120],"function":[104],"give":[109],"bijective":[110],"proofs":[111],"equivalence":[114],"several":[116],"forms":[117],"this":[119],"function.":[121],"bijections":[123],"imply":[124],"new":[128,137],"distribution.":[134],"Using":[135],"statistics,":[138],"we":[139],"conjecture":[140],"certain":[148],"generalizations":[149]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}