scholarly journals Several Explicit and Recursive Formulas for the Generalized Motzkin Numbers

2017 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Recursive Formula ◽  
Catalan Numbers ◽  
Explicit Formulas ◽  
Motzkin Numbers ◽  
Recursive Formulas

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.

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 On distances in generalized Sierpiński graphs

2018 ◽  
Vol 12 (1) ◽  
pp. 49-69 ◽  
Author(s):  
Alejandro Estrada-Moreno ◽  
Erick Rodríguez-Bazan ◽  
Juan Rodríguez-Velázquez
Keyword(s):  
Explicit Formula ◽  
Recursive Formula ◽  
Arbitrary Vertex ◽  
Base Graph ◽  
Sierpiński Graphs ◽  
Recursive Formulas

In this paper we propose formulas for the distance between vertices of a generalized Sierpi?ski graph S(G, t) in terms of the distance between vertices of the base graph G. In particular, we deduce a recursive formula for the distance between an arbitrary vertex and an extreme vertex of S(G, t), and we obtain a recursive formula for the distance between two arbitrary vertices of S(G, t) when the base graph is triangle-free. From these recursive formulas, we provide algorithms to compute the distance between vertices of S(G, t). In addition, we give an explicit formula for the diameter and radius of S(G, t) when the base graph is a tree.

scholarly journals On the Stability of Recursive Formulas

1993 ◽  
Vol 23 (2) ◽  
pp. 227-258 ◽  
Author(s):  
Harry H. Panjer ◽  
Shaun Wang
Keyword(s):  
Binomial Distribution ◽  
Negative Binomial ◽  
Recursive Formula ◽  
Absolute Error ◽  
Recurrence Equation ◽  
Simple Method ◽  
Numerical Examples ◽  
Compound Binomial ◽  
Recursive Formulas ◽  
The Stability

AbstractBased on recurrence equation theory and relative error (rather than absolute error) analysis, the concept and criterion for the stability of a recurrence equation are clarified. A family of recursions, called congruent recursions, is proved to be strongly stable in evaluating its non-negative solutions. A type of strongly unstable recursion is identified. The recursive formula discussed by Panjer (1981) is proved to be strongly stable in evaluating the compound Poisson and the compound Negative Binomial (including Geometric) distributions. For the compound Binomial distribution, the recursion is shown to be unstable. A simple method to cope with this instability is proposed. Many other recursions are reviewed. Illustrative numerical examples are given.

Exchanges of "Quanta" Among Distinguishable and Indistinguishable Objects

1997 ◽  
Vol 11 (19) ◽  
pp. 2281-2301
Author(s):  
Antonio Bonelli ◽  
Stefano Ruffo
Keyword(s):  
Asymptotic Behavior ◽  
Recursion Relation ◽  
Combinatorial Problems ◽  
Physical Problem ◽  
Explicit Formulas ◽  
Recursive Solution ◽  
Theoretic Problem ◽  
Recursive Formulas ◽  
Bose Einstein ◽  
Positive Integers

Beginning with a physical problem of exchange of n indistinguishable "quanta" of energy in an ensemble of k oscillators we define a new wide class of combinatorial problems, which also contains statistics intermediate between Fermi–Dirac and Bose–Einstein. One such problem is related to the number theoretic problem of computing the partitions of positive integers. After establishing such a connection, we give explicit formulas for the partitions M(n,k) of an integer n into k parts with k ≤ 4. Moreover, we derive a recursion relation for fixed n and varying k which is valid for any k. A formula which relates partitions to the cardinality of the partition set taking order into account is also derived. The leading asymptotic behavior for n large is obtained for any k. A suggestive interpretation of this formulas is proposed in terms of simplicial lattices. Recursive formulas at fixed k and varying n are then written for k ≤ 5 using the concept of factorial triangle, which is amenable for generalizations to larger k's. The problem of distinct partitions is mapped onto the probability problem of ball removal with replacement, for which we give again recursive solution formulas. Finally, the method of generalized Tartaglia triangle allows the derivation of recursive formulas for limited partitions which take order into account. This latter result is related to the problem of finding the number of distinct ways of dividing n indistinguishable objects into k distinguishable groups, for which explicit summations had been previously found.

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.

MARGIN TRADING THROUGH HYPER TIMELINE

2007 ◽  
Vol 10 (05) ◽  
pp. 801-815
Author(s):  
SIU-AH NG
Keyword(s):  
Nonstandard Analysis ◽  
Trading Strategy ◽  
Catalan Numbers ◽  
Independent Interest ◽  
Barrier Option ◽  
Explicit Formulas

We consider a model of margin trading based on the hyperfinite timeline. Using only elementary nonstandard analysis we are able to derive explicit formulas for the expected margin call time and loss. Further margin trading strategy is studied and an application to pricing barrier option is given. We prove a generalization of the Catalan numbers which forms the combinatoric basis of our results and should be of independent interest.

EXACT p-ADIC ORDERS FOR DIFFERENCES OF MOTZKIN NUMBERS

2014 ◽  
Vol 10 (03) ◽  
pp. 653-667 ◽  
Author(s):  
TAMÁS LENGYEL
Keyword(s):  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Motzkin Numbers ◽  
Modulo P ◽  
The Difference ◽  
Covering Systems

For any prime p, we establish congruences modulo pn+1 for the difference of the pn+1 th and pn th Motzkin numbers and determine the p-adic order of the difference. The results confirm recent conjectures on the order. The applied techniques involve the use of congruences for the differences of certain Catalan numbers and binomial coefficients, congruential identities for sums of Catalan numbers, central binomial and trinomial coefficients, infinite incongruent disjoint covering systems and the solution of congruential recurrences.

scholarly journals Characteristic Polynomials and Eigenvalues for Certain Classes of Pentadiagonal Matrices

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1056
Author(s):  
María Alejandra Alvarez ◽  
André Ebling Brondani ◽  
Francisca Andrea Macedo França ◽  
Luis A. Medina C.
Keyword(s):  
Characteristic Polynomial ◽  
Diagonal Matrix ◽  
Symmetric Matrices ◽  
Characteristic Polynomials ◽  
Explicit Formulas ◽  
Recursive Formulas

There exist pentadiagonal matrices which are diagonally similar to symmetric matrices. In this work we describe explicitly the diagonal matrix that gives this transformation for certain pentadiagonal matrices. We also consider particular classes of pentadiagonal matrices and obtain recursive formulas for the characteristic polynomial and explicit formulas for their eigenvalues.

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