{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/64609"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/64609","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Combinatorial enumeration of weighted Catalan numbers","abstract":"This thesis is devoted to the divisibility property of weighted Catalan and Motzkin numbers and its applications. In Chapter 1, the definitions and properties of weighted Catalan and Motzkin numbers are introduced. Chapter 2 studies Wilf conjecture on the complementary Bell number, the alternating sum of the Stirling number of the second kind. Congruence properties of the complementary Bell numbers are found by weighted Motkin paths, and Wilf conjecture is partially proved. In Chapter 3, Konvalinka conjecture is proved. It is a conjecture on the largest power of two dividing weighted Catalan number, when the weight function is a polynomial. As a corollary, we provide another proof of Postnikov and Sagan of weighted Catalan numbers, and we also generalize Konvalinka conjecture for a general weight function.","abstract_html":"This thesis is devoted to the divisibility property of weighted Catalan and Motzkin numbers and its applications. In Chapter 1, the definitions and properties of weighted Catalan and Motzkin numbers are introduced. Chapter 2 studies Wilf conjecture on the complementary Bell number, the alternating sum of the Stirling number of the second kind. Congruence properties of the complementary Bell numbers are found by weighted Motkin paths, and Wilf conjecture is partially proved. In Chapter 3, Konvalinka conjecture is proved. It is a conjecture on the largest power of two dividing weighted Catalan number, when the weight function is a polynomial. As a corollary, we provide another proof of Postnikov and Sagan of weighted Catalan numbers, and we also generalize Konvalinka conjecture for a general weight function.","abstract_has_math":false,"creators":["An, Junkyu"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Dept. of Mathematics.","school":null,"contributors":[],"advisors":["Alexander Postnikov."],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010","date_published":"2010","updated_at":"2026-07-22T22:22:07Z","subjects":["Mathematics."],"languages":["eng"],"rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1721.1/64609","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Alexander Postnikov."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Dept. of Mathematics."]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Massachusetts Institute of Technology. Dept. of Mathematics."]},{"key":"dc:creator","label":"Author","values":["An, Junkyu"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2011-06-20T15:59:46Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2011-06-20T15:59:46Z"]},{"key":"dc:date.issued","label":"Date","values":["2010"]},{"key":"dc:publisher","label":"Institution","values":["Massachusetts Institute of Technology"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics."]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://dspace.mit.edu/handle/1721.1/7582"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1721.1/64609"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.","Cataloged from PDF version of thesis.","Includes bibliographical references (p. 69-70)."]},{"key":"dc:description.abstract","label":"Abstract","values":["This thesis is devoted to the divisibility property of weighted Catalan and Motzkin numbers and its applications. In Chapter 1, the definitions and properties of weighted Catalan and Motzkin numbers are introduced. Chapter 2 studies Wilf conjecture on the complementary Bell number, the alternating sum of the Stirling number of the second kind. Congruence properties of the complementary Bell numbers are found by weighted Motkin paths, and Wilf conjecture is partially proved. In Chapter 3, Konvalinka conjecture is proved. It is a conjecture on the largest power of two dividing weighted Catalan number, when the weight function is a polynomial. As a corollary, we provide another proof of Postnikov and Sagan of weighted Catalan numbers, and we also generalize Konvalinka conjecture for a general weight function."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Combinatorial enumeration of weighted Catalan numbers"]}]}],"canonical_facts":{"dc:contributor.advisor":["Alexander Postnikov."],"dc:contributor.department":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:contributor.other":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:creator":["An, Junkyu"],"dc:date.accessioned":["2011-06-20T15:59:46Z"],"dc:date.available":["2011-06-20T15:59:46Z"],"dc:date.issued":["2010"],"dc:description":["Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.","Cataloged from PDF version of thesis.","Includes bibliographical references (p. 69-70)."],"dc:description.abstract":["This thesis is devoted to the divisibility property of weighted Catalan and Motzkin numbers and its applications. In Chapter 1, the definitions and properties of weighted Catalan and Motzkin numbers are introduced. Chapter 2 studies Wilf conjecture on the complementary Bell number, the alternating sum of the Stirling number of the second kind. Congruence properties of the complementary Bell numbers are found by weighted Motkin paths, and Wilf conjecture is partially proved. In Chapter 3, Konvalinka conjecture is proved. It is a conjecture on the largest power of two dividing weighted Catalan number, when the weight function is a polynomial. As a corollary, we provide another proof of Postnikov and Sagan of weighted Catalan numbers, and we also generalize Konvalinka conjecture for a general weight function."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["http://hdl.handle.net/1721.1/64609"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Mathematics."],"dc:title":["Combinatorial enumeration of weighted Catalan numbers"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:22:07Z"}