2026-09-19T05:14:19Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/646092022-01-13T07:54:35Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Alexander Postnikov.
An, Junkyu
Massachusetts Institute of Technology. Dept. of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2011-06-20T15:59:46Z
2011-06-20T15:59:46Z
2010
2010
http://hdl.handle.net/1721.1/64609
727153781
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 69-70).
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.
by Junkyu An.
Ph.D.
70 p.
eng
Massachusetts Institute of Technology
M.I.T. theses are protected by
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http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Combinatorial enumeration of weighted Catalan numbers
Thesis
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Publication
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<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>
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