{"id":"https://openalex.org/W2143292021","doi":"https://doi.org/10.37236/1709","title":"A Schr\u00f6der Generalization of Haglund's Statistic on Catalan Paths","display_name":"A Schr\u00f6der Generalization of Haglund's Statistic on Catalan Paths","publication_year":2003,"publication_date":"2003-04-23","ids":{"openalex":"https://openalex.org/W2143292021","doi":"https://doi.org/10.37236/1709","mag":"2143292021"},"language":"en","primary_location":{"id":"doi:10.37236/1709","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1709","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v10i1r16/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":false,"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":"bronze","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v10i1r16/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5078112701","display_name":"Eric S. Egge","orcid":null},"institutions":[{"id":"https://openalex.org/I100191712","display_name":"Pepperdine University","ror":"https://ror.org/0529ybh43","country_code":"US","type":"education","lineage":["https://openalex.org/I100191712"]},{"id":"https://openalex.org/I176113532","display_name":"Gettysburg College","ror":"https://ror.org/045e26x92","country_code":"US","type":"education","lineage":["https://openalex.org/I176113532"]},{"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":"E. S. Egge","raw_affiliation_strings":["Department of Mathematics Gettysburg College, Gettysburg, PA 17325","Department of Mathematics University of Pennsylvania, Philadelphia, PA 19104","Mathematics Department Pepperdine University, Malibu, CA 90263-4321"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics Gettysburg College, Gettysburg, PA 17325","institution_ids":["https://openalex.org/I176113532"]},{"raw_affiliation_string":"Department of Mathematics University of Pennsylvania, Philadelphia, PA 19104","institution_ids":["https://openalex.org/I79576946"]},{"raw_affiliation_string":"Mathematics Department Pepperdine University, Malibu, CA 90263-4321","institution_ids":["https://openalex.org/I100191712"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5110172039","display_name":"J. Haglund","orcid":null},"institutions":[{"id":"https://openalex.org/I100191712","display_name":"Pepperdine University","ror":"https://ror.org/0529ybh43","country_code":"US","type":"education","lineage":["https://openalex.org/I100191712"]},{"id":"https://openalex.org/I176113532","display_name":"Gettysburg College","ror":"https://ror.org/045e26x92","country_code":"US","type":"education","lineage":["https://openalex.org/I176113532"]},{"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":"J. Haglund","raw_affiliation_strings":["Department of Mathematics Gettysburg College, Gettysburg, PA 17325","Department of Mathematics University of Pennsylvania, Philadelphia, PA 19104","Mathematics Department Pepperdine University, Malibu, CA 90263-4321"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics Gettysburg College, Gettysburg, PA 17325","institution_ids":["https://openalex.org/I176113532"]},{"raw_affiliation_string":"Department of Mathematics University of Pennsylvania, Philadelphia, PA 19104","institution_ids":["https://openalex.org/I79576946"]},{"raw_affiliation_string":"Mathematics Department Pepperdine University, Malibu, CA 90263-4321","institution_ids":["https://openalex.org/I100191712"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5080635755","display_name":"Kendra Killpatrick","orcid":"https://orcid.org/0000-0002-9608-5605"},"institutions":[{"id":"https://openalex.org/I100191712","display_name":"Pepperdine University","ror":"https://ror.org/0529ybh43","country_code":"US","type":"education","lineage":["https://openalex.org/I100191712"]},{"id":"https://openalex.org/I176113532","display_name":"Gettysburg College","ror":"https://ror.org/045e26x92","country_code":"US","type":"education","lineage":["https://openalex.org/I176113532"]},{"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":"K. Killpatrick","raw_affiliation_strings":["Department of Mathematics Gettysburg College, Gettysburg, PA 17325","Department of Mathematics University of Pennsylvania, Philadelphia, PA 19104","Mathematics Department Pepperdine University, Malibu, CA 90263-4321"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics Gettysburg College, Gettysburg, PA 17325","institution_ids":["https://openalex.org/I176113532"]},{"raw_affiliation_string":"Department of Mathematics University of Pennsylvania, Philadelphia, PA 19104","institution_ids":["https://openalex.org/I79576946"]},{"raw_affiliation_string":"Mathematics Department Pepperdine University, Malibu, CA 90263-4321","institution_ids":["https://openalex.org/I100191712"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5080202943","display_name":"Dirk Kremer","orcid":null},"institutions":[{"id":"https://openalex.org/I100191712","display_name":"Pepperdine University","ror":"https://ror.org/0529ybh43","country_code":"US","type":"education","lineage":["https://openalex.org/I100191712"]},{"id":"https://openalex.org/I176113532","display_name":"Gettysburg College","ror":"https://ror.org/045e26x92","country_code":"US","type":"education","lineage":["https://openalex.org/I176113532"]},{"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":"D. Kremer","raw_affiliation_strings":["Department of Mathematics Gettysburg College, Gettysburg, PA 17325","Department of Mathematics University of Pennsylvania, Philadelphia, PA 19104","Mathematics Department Pepperdine University, Malibu, CA 90263-4321"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics Gettysburg College, Gettysburg, PA 17325","institution_ids":["https://openalex.org/I176113532"]},{"raw_affiliation_string":"Department of Mathematics University of Pennsylvania, Philadelphia, PA 19104","institution_ids":["https://openalex.org/I79576946"]},{"raw_affiliation_string":"Mathematics Department Pepperdine University, Malibu, CA 90263-4321","institution_ids":["https://openalex.org/I100191712"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.6092,"has_fulltext":true,"cited_by_count":27,"citation_normalized_percentile":{"value":0.81439394,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"10","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":0.9994000196456909,"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":0.9994000196456909,"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":0.9908999800682068,"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/T13506","display_name":"Botanical Research and Chemistry","score":0.9641000032424927,"subfield":{"id":"https://openalex.org/subfields/1105","display_name":"Ecology, Evolution, Behavior and Systematics"},"field":{"id":"https://openalex.org/fields/11","display_name":"Agricultural and Biological Sciences"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.8223052620887756},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8071425557136536},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7896371483802795},{"id":"https://openalex.org/keywords/catalan-number","display_name":"Catalan number","score":0.569019079208374},{"id":"https://openalex.org/keywords/lattice","display_name":"Lattice (music)","score":0.5564977526664734},{"id":"https://openalex.org/keywords/symmetric-function","display_name":"Symmetric function","score":0.5464680194854736},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.46147724986076355},{"id":"https://openalex.org/keywords/symmetric-polynomial","display_name":"Symmetric polynomial","score":0.4475814998149872},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.44178566336631775},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.43342870473861694},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3355468809604645},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.11364391446113586},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.09533590078353882}],"concepts":[{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.8223052620887756},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8071425557136536},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7896371483802795},{"id":"https://openalex.org/C43929256","wikidata":"https://www.wikidata.org/wiki/Q270513","display_name":"Catalan number","level":2,"score":0.569019079208374},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.5564977526664734},{"id":"https://openalex.org/C3746008","wikidata":"https://www.wikidata.org/wiki/Q981351","display_name":"Symmetric function","level":2,"score":0.5464680194854736},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.46147724986076355},{"id":"https://openalex.org/C52153879","wikidata":"https://www.wikidata.org/wiki/Q930499","display_name":"Symmetric polynomial","level":4,"score":0.4475814998149872},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.44178566336631775},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.43342870473861694},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3355468809604645},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11364391446113586},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.09533590078353882},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.37236/1709","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1709","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v10i1r16/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":false,"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:10.1.1.211.192","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.211.192","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.upenn.edu/%7Ejhaglund/preprints/qtsch.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/1709","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1709","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v10i1r16/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":false,"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":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2143292021.pdf","grobid_xml":"https://content.openalex.org/works/W2143292021.grobid-xml"},"referenced_works_count":19,"referenced_works":["https://openalex.org/W1634042400","https://openalex.org/W1671066123","https://openalex.org/W2008052995","https://openalex.org/W2022892305","https://openalex.org/W2042843318","https://openalex.org/W2046479912","https://openalex.org/W2048202079","https://openalex.org/W2059036144","https://openalex.org/W2087941871","https://openalex.org/W2096502393","https://openalex.org/W2134984950","https://openalex.org/W2140141642","https://openalex.org/W2327812169","https://openalex.org/W2914659449","https://openalex.org/W2914782162","https://openalex.org/W3136846898","https://openalex.org/W4239240946","https://openalex.org/W4300302225","https://openalex.org/W4384817650"],"related_works":["https://openalex.org/W2964230509","https://openalex.org/W2785232145","https://openalex.org/W4226464148","https://openalex.org/W3013709019","https://openalex.org/W4282813460","https://openalex.org/W4381712798","https://openalex.org/W1560962904","https://openalex.org/W641273746","https://openalex.org/W2093479061","https://openalex.org/W2901510162"],"abstract_inverted_index":{"Garsia":[0,39],"and":[1,37,66,69,95,113,119,138],"Haiman":[2],"(J.":[3],"Algebraic.":[4],"Combin.":[5],"$\\bf5$":[6],"$(1996)$,":[7],"$191-244$)":[8],"conjectured":[9],"that":[10],"a":[11,22,129],"certain":[12,105],"sum":[13,59],"$C_n(q,t)$":[14],"of":[15,77,87,107,145],"rational":[16,134],"functions":[17,135],"in":[18,24,35,75,124,136],"$q,t$":[19,25,137],"reduces":[20],"to":[21,91],"polynomial":[23],"with":[26,38],"nonnegative":[27],"integral":[28],"coefficients.":[29],"Haglund":[30],"later":[31],"discovered":[32],"(Adv.":[33],"Math.,":[34],"press),":[36],"proved":[40],"(Proc.":[41],"Nat.":[42],"Acad.":[43],"Sci.":[44],"${\\bf98}$":[45],"$(2001)$,":[46],"$4313-4316$)":[47],"the":[48,58,78,139,143],"refined":[49],"conjecture":[50,120,132],"$C_n(q,t)":[51],"=":[52],"\\sum":[53],"q^{{\\rm":[54],"area}}t^{{\\rm":[55,102],"bounce}}$.":[56],"Here":[57],"is":[60],"over":[61,104],"all":[62],"Catalan":[63],"lattice":[64,93],"paths":[65],"${\\rm":[67,70],"area}$":[68],"bounce}$":[71],"have":[72],"simple":[73],"descriptions":[74],"terms":[76],"path.":[79],"In":[80],"this":[81],"article":[82],"we":[83],"give":[84],"an":[85],"extension":[86],"$({\\rm":[88],"area},{\\rm":[89],"bounce})$":[90],"Schr\u00f6der":[92,108],"paths,":[94],"introduce":[96],"polynomials":[97],"defined":[98],"by":[99],"summing":[100],"$q^{{\\rm":[101],"bounce}}$":[103],"sets":[106],"paths.":[109],"We":[110,126],"derive":[111],"recurrences":[112],"special":[114],"values":[115],"for":[116],"these":[117],"polynomials,":[118],"they":[121],"are":[122],"symmetric":[123,147],"$q,t$.":[125],"also":[127],"describe":[128],"much":[130],"stronger":[131],"involving":[133],"$\\nabla$":[140],"operator":[141],"from":[142],"theory":[144],"Macdonald":[146],"functions.":[148]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":5}],"updated_date":"2026-08-26T07:47:46.906454","created_date":"2025-10-10T00:00:00"}
