{"id":"https://openalex.org/W3197609398","doi":"https://doi.org/10.3934/amc.2021032","title":"On the number of distinct <i>k</i>-decks: Enumeration and bounds","display_name":"On the number of distinct <i>k</i>-decks: Enumeration and bounds","publication_year":2021,"publication_date":"2021-09-08","ids":{"openalex":"https://openalex.org/W3197609398","doi":"https://doi.org/10.3934/amc.2021032","mag":"3197609398"},"language":"en","primary_location":{"id":"doi:10.3934/amc.2021032","is_oa":true,"landing_page_url":"https://doi.org/10.3934/amc.2021032","pdf_url":"https://www.aimsciences.org/data/article/export-pdf?id=5ffbb1f4-c91f-487f-84af-f4e4b16134e4","source":{"id":"https://openalex.org/S161492159","display_name":"Advances in Mathematics of Communications","issn_l":"1930-5338","issn":["1930-5338","1930-5346"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Mathematics of Communications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.aimsciences.org/data/article/export-pdf?id=5ffbb1f4-c91f-487f-84af-f4e4b16134e4","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5014193799","display_name":"Johan Chrisnata","orcid":"https://orcid.org/0000-0003-1705-9597"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Johan Chrisnata","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5083157397","display_name":"Han Mao Kiah","orcid":"https://orcid.org/0000-0001-5611-0848"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Han Mao Kiah","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5074078078","display_name":"Sankeerth Rao Karingula","orcid":"https://orcid.org/0000-0003-2212-4322"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Sankeerth Rao Karingula","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5004102644","display_name":"Alexander Vardy","orcid":"https://orcid.org/0000-0003-3303-9078"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Alexander Vardy","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5109852840","display_name":"E. Y. Yao","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Eitan Yaakobi Yao","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5010924440","display_name":"Hanwen Yao","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Hanwen Yao","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5014193799"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.4079,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.68802491,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"17","issue":"4","first_page":"960","last_page":"978"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9994000196456909,"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"}},"topics":[{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9994000196456909,"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"}},{"id":"https://openalex.org/T11269","display_name":"Algorithms and Data Compression","score":0.9993000030517578,"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"}},{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/mathematics","display_name":"Mathematics","score":0.8770744800567627},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8577229976654053},{"id":"https://openalex.org/keywords/enumeration","display_name":"Enumeration","score":0.7623677253723145},{"id":"https://openalex.org/keywords/multiset","display_name":"Multiset","score":0.7411860227584839},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.7279386520385742},{"id":"https://openalex.org/keywords/alphabet","display_name":"Alphabet","score":0.6796582937240601},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.6283423900604248},{"id":"https://openalex.org/keywords/binary-number","display_name":"Binary number","score":0.580737292766571},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3540632724761963},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.14148977398872375},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.09857919812202454},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.09303665161132812}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8770744800567627},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8577229976654053},{"id":"https://openalex.org/C156340839","wikidata":"https://www.wikidata.org/wiki/Q2704791","display_name":"Enumeration","level":2,"score":0.7623677253723145},{"id":"https://openalex.org/C2779623528","wikidata":"https://www.wikidata.org/wiki/Q864377","display_name":"Multiset","level":2,"score":0.7411860227584839},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.7279386520385742},{"id":"https://openalex.org/C112876837","wikidata":"https://www.wikidata.org/wiki/Q837518","display_name":"Alphabet","level":2,"score":0.6796582937240601},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.6283423900604248},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.580737292766571},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3540632724761963},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.14148977398872375},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.09857919812202454},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.09303665161132812},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3934/amc.2021032","is_oa":true,"landing_page_url":"https://doi.org/10.3934/amc.2021032","pdf_url":"https://www.aimsciences.org/data/article/export-pdf?id=5ffbb1f4-c91f-487f-84af-f4e4b16134e4","source":{"id":"https://openalex.org/S161492159","display_name":"Advances in Mathematics of Communications","issn_l":"1930-5338","issn":["1930-5338","1930-5346"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Mathematics of Communications","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.3934/amc.2021032","is_oa":true,"landing_page_url":"https://doi.org/10.3934/amc.2021032","pdf_url":"https://www.aimsciences.org/data/article/export-pdf?id=5ffbb1f4-c91f-487f-84af-f4e4b16134e4","source":{"id":"https://openalex.org/S161492159","display_name":"Advances in Mathematics of Communications","issn_l":"1930-5338","issn":["1930-5338","1930-5346"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Mathematics of Communications","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3197609398.pdf","grobid_xml":"https://content.openalex.org/works/W3197609398.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W137071685","https://openalex.org/W2014342457","https://openalex.org/W2031816626","https://openalex.org/W2058068142","https://openalex.org/W2219765273","https://openalex.org/W2336019055","https://openalex.org/W2468104078","https://openalex.org/W2584062640","https://openalex.org/W2912739023","https://openalex.org/W2963412269","https://openalex.org/W2963545007","https://openalex.org/W2963561675","https://openalex.org/W2969308577","https://openalex.org/W2989916841","https://openalex.org/W3004384165","https://openalex.org/W3027715556"],"related_works":["https://openalex.org/W4300455649","https://openalex.org/W2296588302","https://openalex.org/W4252409532","https://openalex.org/W4328119206","https://openalex.org/W2033245919","https://openalex.org/W108845024","https://openalex.org/W2006067547","https://openalex.org/W9228044","https://openalex.org/W3143753676","https://openalex.org/W2950705774"],"abstract_inverted_index":{"The":[0],"$":[1,18,22,24,30,38,50,52,57,59,61,69,71,73,75,80,90,92,97,103,105,107,111,113,128,145,149,151,157,166,170,176,178,180],"k":[2,19,31,58,74,81,108,114,158],"$-deck":[3],"of":[4,12,16,28,36,49,56,68],"a":[5,85,133],"sequence":[6],"is":[7,76,179],"defined":[8],"as":[9],"the":[10,26,46,123,172],"multiset":[11],"all":[13],"its":[14],"subsequences":[15],"length":[17,37],"$.":[20,40,99,119,155,182],"Let":[21],"D_k(n)":[23,51,70,91,104,129,146,152],"denote":[25],"number":[27],"distinct":[29],"$-decks":[32],"for":[33,53,78,106,139],"binary":[34,42,140],"sequences":[35],"n":[39,62,98,116],"For":[41,156],"alphabet,":[43,141],"we":[44,83,142,162],"determine":[45],"exact":[47],"value":[48],"small":[54],"values":[55],"and":[60,64,112,131,150],"$,":[63,82,130,161],"provide":[65,132],"asymptotic":[66,124],"estimates":[67],"when":[72],"fixed.Specifically,":[77],"fixed":[79],"introduce":[84],"trellis-based":[86],"method":[87],"to":[88],"compute":[89,102],"in":[93,96],"time":[94],"polynomial":[95],"We":[100,120],"then":[101],"\\in":[109],"\\{3,4,5,6\\}":[110],"\\leqslant":[115,117],"30":[118],"also":[121],"improve":[122],"upper":[125,173],"bound":[126,135,174],"on":[127,175],"lower":[134],"thereupon.":[136],"In":[137],"particular,":[138],"show":[143,164],"that":[144,165],"=":[147,153,159,168],"O\\bigl(n^{(k-1)2^{k-1}+1}\\bigr)":[148],"\\Omega(n^k)":[154],"3":[160],"moreover":[163],"D_3(n)":[167,177],"\\Omega(n^6)":[169],"while":[171],"O(n^9)":[181]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1}],"updated_date":"2026-01-13T01:12:25.745995","created_date":"2025-10-10T00:00:00"}