scholarly journals Motzkin numbers

1977 ◽  
Vol 23 (3) ◽  
pp. 291-301 ◽  
Author(s):  
Robert Donaghey ◽  
Louis W Shapiro
Keyword(s):  
Motzkin Numbers

scholarly journals Hankel determinants of sums of consecutive Motzkin numbers

2011 ◽  
Vol 434 (3) ◽  
pp. 712-722 ◽  
Author(s):  
Naiomi T. Cameron ◽  
Andrew C.M. Yip
Keyword(s):  
Hankel Determinants ◽  
Motzkin Numbers

Vexillary Involutions are Enumerated by Motzkin Numbers

2001 ◽  
Vol 5 (2) ◽  
pp. 153-174 ◽  
Author(s):  
O. Guibert ◽  
E. Pergola ◽  
R. Pinzani
Keyword(s):  
Motzkin Numbers

scholarly journals Congruences for Catalan and Motzkin numbers and related sequences

2006 ◽  
Vol 117 (1) ◽  
pp. 191-215 ◽  
Author(s):  
Emeric Deutsch ◽  
Bruce E. Sagan
Keyword(s):  
Motzkin Numbers

scholarly journals Taylor expansions for Catalan and Motzkin numbers

2002 ◽  
Vol 29 (3) ◽  
pp. 345-357 ◽  
Author(s):  
Sen-Peng Eu ◽  
Shu-Chung Liu ◽  
Yeong-Nan Yeh
Keyword(s):  
Taylor Expansions ◽  
Motzkin Numbers

scholarly journals A Classification of Motzkin Numbers Modulo 8

10.37236/7092 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Ying Wang ◽  
Guoce Xin
Keyword(s):  
Short Proof ◽  
Recursive Formula ◽  
Catalan Numbers ◽  
Motzkin Numbers ◽  
Full Classification ◽  
Factorial Representation

The well-known Motzkin numbers were conjectured by Deutsch and Sagan to be nonzero when modulo $8$. The conjecture was first proved by Sen-Peng Eu, Shu-chung Liu and Yeong-Nan Yeh by using the factorial representation of the Catalan numbers. We present a short proof by finding a recursive formula for Motzkin numbers modulo $8$. Moreover, such a recursion leads to a full classification of Motzkin numbers modulo $8$. An addendum was added on April 3 2018.

scholarly journals Several explicit and recursive formulas for generalized Motzkin numbers

2020 ◽  
Vol 5 (2) ◽  
pp. 1333-1345
Author(s):  
Feng Qi ◽  
◽  
Bai-Ni Guo ◽  
◽  
Keyword(s):  
Motzkin Numbers ◽  
Recursive Formulas

scholarly journals A Bijection on Dyck Paths and its Cycle Structure

10.37236/946 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
David Callan
Keyword(s):  
Catalan Numbers ◽  
Complete Analysis ◽  
Cycle Structure ◽  
Pascal Matrix ◽  
Dyck Paths ◽  
Motzkin Numbers

The known bijections on Dyck paths are either involutions or have notoriously intractable cycle structure. Here we present a size-preserving bijection on Dyck paths whose cycle structure is amenable to complete analysis. In particular, each cycle has length a power of 2. A new manifestation of the Catalan numbers as labeled forests crops up en route as does the Pascal matrix mod 2. We use the bijection to show the equivalence of two known manifestations of the Motzkin numbers. Finally, we consider some statistics on the new Catalan manifestation and the identities they interpret.

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