{"id":"https://openalex.org/W1613508308","doi":"https://doi.org/10.37236/200","title":"$q$-Counting Descent Pairs with Prescribed Tops and Bottoms","display_name":"$q$-Counting Descent Pairs with Prescribed Tops and Bottoms","publication_year":2009,"publication_date":"2009-08-31","ids":{"openalex":"https://openalex.org/W1613508308","doi":"https://doi.org/10.37236/200","mag":"1613508308"},"language":"en","primary_location":{"id":"doi:10.37236/200","is_oa":true,"landing_page_url":"https://doi.org/10.37236/200","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r111/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/v16i1r111/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101456870","display_name":"John Hall","orcid":null},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"John Hall","raw_affiliation_strings":["Department of Mathematics, Harvard University, Cambridge, MA,"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Harvard University, Cambridge, MA,","institution_ids":["https://openalex.org/I136199984"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5021271761","display_name":"Jeffrey Liese","orcid":"https://orcid.org/0000-0001-7538-2945"},"institutions":[{"id":"https://openalex.org/I149919469","display_name":"California Polytechnic State University","ror":"https://ror.org/001gpfp45","country_code":"US","type":"education","lineage":["https://openalex.org/I149919469"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jeffrey Liese","raw_affiliation_strings":["Department of Mathematics, California Polytechnic State University, San Luis Obispo, CA 93407,"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, California Polytechnic State University, San Luis Obispo, CA 93407,","institution_ids":["https://openalex.org/I149919469"]}]},{"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"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jeffrey B. Remmel","raw_affiliation_strings":["Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, jrem-"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, jrem-","institution_ids":["https://openalex.org/I36258959"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5101456870"],"corresponding_institution_ids":["https://openalex.org/I136199984"],"apc_list":null,"apc_paid":null,"fwci":0.4734,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.53213545,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"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":0.9990000128746033,"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":0.9990000128746033,"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":0.9954000115394592,"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/T11901","display_name":"Bayesian Methods and Mixture Models","score":0.9921000003814697,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/sigma","display_name":"Sigma","score":0.8418523073196411},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8279812932014465},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6933168172836304},{"id":"https://openalex.org/keywords/descent","display_name":"Descent (aeronautics)","score":0.6188362240791321},{"id":"https://openalex.org/keywords/permutation","display_name":"Permutation (music)","score":0.614359438419342},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.12362554669380188},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.0443805456161499}],"concepts":[{"id":"https://openalex.org/C2778049214","wikidata":"https://www.wikidata.org/wiki/Q7512234","display_name":"Sigma","level":2,"score":0.8418523073196411},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8279812932014465},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6933168172836304},{"id":"https://openalex.org/C2776637919","wikidata":"https://www.wikidata.org/wiki/Q624380","display_name":"Descent (aeronautics)","level":2,"score":0.6188362240791321},{"id":"https://openalex.org/C21308566","wikidata":"https://www.wikidata.org/wiki/Q7169365","display_name":"Permutation (music)","level":2,"score":0.614359438419342},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.12362554669380188},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0443805456161499},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C153294291","wikidata":"https://www.wikidata.org/wiki/Q25261","display_name":"Meteorology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/200","is_oa":true,"landing_page_url":"https://doi.org/10.37236/200","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r111/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:10.37236/200","is_oa":true,"landing_page_url":"https://doi.org/10.37236/200","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r111/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/G4845796323","display_name":null,"funder_award_id":"DMS 0654060","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6176407360","display_name":"Combinatorial Structures for Permutation Enumeration and Macdonald Polynomials","funder_award_id":"0654060","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/W1613508308.pdf","grobid_xml":"https://content.openalex.org/works/W1613508308.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W2040956452","https://openalex.org/W2049647979","https://openalex.org/W2914782162"],"related_works":["https://openalex.org/W2486267010","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W1678820847","https://openalex.org/W4245490552","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2949219549","https://openalex.org/W4287905741","https://openalex.org/W4280528977"],"abstract_inverted_index":{"Given":[0],"sets":[1],"$X$":[2,35],"and":[3,8,36,45,77],"$Y$":[4],"of":[5,21,54,84,103],"positive":[6],"integers":[7],"a":[9,24,69],"permutation":[10],"$\\sigma":[11,55,89],"=":[12],"\\sigma_1":[13],"\\sigma_2":[14],"\\cdots":[15],"\\sigma_n":[16],"\\in":[17,56,90],"S_n$,":[18],"an":[19],"$(X,Y)$-descent":[20],"$\\sigma$":[22,76],"is":[23,33,40],"descent":[25],"pair":[26],"$\\sigma_i":[27],">":[28],"\\sigma_{i+1}$":[29],"whose":[30,37],"\"top\"":[31],"$\\sigma_i$":[32],"in":[34,41,65],"\"bottom\"":[38],"$\\sigma_{i+1}$":[39],"$Y$.":[42],"Recently":[43],"Hall":[44],"Remmel":[46],"proved":[47],"two":[48],"formulas":[49,106],"for":[50,107],"the":[51,82,104],"number":[52],"$P_{n,s}^{X,Y}$":[53],"S_n$":[57,91],"with":[58,92],"$s$":[59,93],"$(X,Y)$-descents,":[60],"which":[61],"generalized":[62],"Liese's":[63],"results":[64],"[1].":[66],"We":[67,95],"define":[68,78],"new":[70],"statistic":[71],"${\\rm":[72],"stat}_{X,Y}(\\sigma)$":[73],"on":[74],"permutations":[75],"$P_{n,s}^{X,Y}(q)$":[79],"to":[80],"be":[81],"sum":[83],"$q^{{\\rm":[85],"stat}_{X,Y}(\\sigma)}$":[86],"over":[87],"all":[88],"$(X,Y)$-descents.":[94],"then":[96],"show":[97],"that":[98],"there":[99],"are":[100],"natural":[101],"$q$-analogues":[102],"Hall-Remmel":[105],"$P_{n,s}^{X,Y}(q)$.":[108]},"counts_by_year":[{"year":2022,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}