motzkin numbers
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Author(s):  
C. Krattenthaler

AbstractWe present a formula that expresses the Hankel determinants of a linear combination of length $$d+1$$ d + 1 of moments of orthogonal polynomials in terms of a $$d\times d$$ d × d determinant of the orthogonal polynomials. This formula exists somehow hidden in the folklore of the theory of orthogonal polynomials but deserves to be better known, and be presented correctly and with full proof. We present four fundamentally different proofs, one that uses classical formulae from the theory of orthogonal polynomials, one that uses a vanishing argument and is due to Elouafi (J Math Anal Appl 431:1253–1274, 2015) (but given in an incomplete form there), one that is inspired by random matrix theory and is due to Brézin and Hikami (Commun Math Phys 214:111–135, 2000), and one that uses (Dodgson) condensation. We give two applications of the formula. In the first application, we explain how to compute such Hankel determinants in a singular case. The second application concerns the linear recurrence of such Hankel determinants for a certain class of moments that covers numerous classical combinatorial sequences, including Catalan numbers, Motzkin numbers, central binomial coefficients, central trinomial coefficients, central Delannoy numbers, Schröder numbers, Riordan numbers, and Fine numbers.


2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Paul M. Rakoyomamonjy ◽  
Sandrataniaina R. Andriantsoa

We examine the enumeration of certain Motzkin objects according to the numbers of crossings and nestings. With respect to continued fractions, we compute and express the distributions of the statistics of the numbers of crossings and nestings over three sets, namely the set of $4321$-avoiding involutions, the set of $3412$-avoiding involutions, and the set of $(321,3\bar{1}42)$-avoiding permutations. To get our results, we exploit the bijection of Biane restricted to the sets of $4321$- and $3412$-avoiding involutions which was characterized by Barnabei et al. and the bijection between $(321,3\bar{1}42)$-avoiding permutations and Motzkin paths, presented by Chen et al.. Furthermore, we manipulate the obtained continued fractions to get the recursion formulas for the polynomial distributions of crossings and nestings, and it follows that the results involve two new $q$-Motzkin numbers.


2020 ◽  
Vol 343 (12) ◽  
pp. 112133
Author(s):  
Xi Chen ◽  
Yi Wang ◽  
Sai-Nan Zheng
Keyword(s):  

2020 ◽  
Vol 5 (2) ◽  
pp. 1333-1345
Author(s):  
Feng Qi ◽  
◽  
Bai-Ni Guo ◽  
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2018 ◽  
Vol 73 ◽  
pp. 114-137
Author(s):  
C. Krattenthaler ◽  
T.W. Müller
Keyword(s):  

10.37236/7092 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Ying Wang ◽  
Guoce Xin

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.


2017 ◽  
Vol 40 (16) ◽  
pp. 5810-5820 ◽  
Author(s):  
Radica Bojičić ◽  
Marko D. Petković ◽  
Predrag M. Rajković

Author(s):  
Feng Qi ◽  
Bai-Ni Guo

In the paper, the authors find two explicit formulas and recover a recursive formula for the generalized Motzkin numbers. Consequently, the authors deduce two explicit formulas and a recursive formula for the Motzkin numbers, the Catalan numbers, and the restricted hexagonal numbers respectively.


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