11 research outputs found
Motzkin numbers and related sequences modulo powers of 2
We show that the generating function for Motzkin
numbers , when coefficients are reduced modulo a given power of , can
be expressed as a polynomial in the basic series with coefficients being Laurent polynomials in and
. We use this result to determine modulo in terms of the binary
digits of~, thus improving, respectively complementing earlier results by
Eu, Liu and Yeh [Europ. J. Combin. 29 (2008), 1449-1466] and by Rowland and
Yassawi [J. Th\'eorie Nombres Bordeaux 27 (2015), 245-288]. Analogous results
are also shown to hold for related combinatorial sequences, namely for the
Motzkin prefix numbers, Riordan numbers, central trinomial coefficients, and
for the sequence of hex tree numbers.Comment: 28 pages, AmS-LaTeX; minor typos correcte
Combinatorial enumeration of weighted Catalan numbers
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
Catalan and Motzkin numbers modulo 4 and 8
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