scholarly journals On the Catalan Numbers and Some of Their Identities

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 62 ◽  
Author(s):  
Wenpeng Zhang ◽  
Li Chen
Keyword(s):  
Catalan Numbers ◽  
Recursive Sequences ◽  
Combinatorial Methods

The main purpose of this paper is using the elementary and combinatorial methods to study the properties of the Catalan numbers, and give two new identities for them. In order to do this, we first introduce two new recursive sequences, then with the help of these sequences, we obtained the identities for the convolution involving the Catalan numbers.

scholarly journals Pattern-Avoiding Set Partitions and Catalan Numbers

10.37236/2048 ◽  
2012 ◽  
Vol 18 (2) ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Kernel Method ◽  
Catalan Numbers ◽  
Set Partitions ◽  
Narayana Numbers ◽  
Combinatorial Methods

We identify several subsets of the partitions of $[n]$, each characterized by the avoidance of a pair of patterns, respectively of lengths four and five.  Each of the classes we consider is enumerated by the Catalan numbers.  Furthermore, the members of each class having a prescribed number of blocks are enumerated by the Narayana numbers.  We use both algebraic and combinatorial methods to establish our results.  In some of the cases, we make use of the kernel method to solve the recurrence arising when a further statistic is considered.  In other cases, we define bijections with previously enumerated classes which preserve the number of blocks.  Two of our bijections are of an algorithmic nature and systematically replace the occurrences of one pattern with those of another having the same length.

scholarly journals A Method for Determining the Mod-$2^k$ Behaviour of Recursive Sequences, with Applications to Subgroup Counting

10.37236/2433 ◽  
2012 ◽  
Vol 18 (2) ◽  
Author(s):  
M. Kauers ◽  
C. Krattenthaler ◽  
T. W. Müller
Keyword(s):  
Finite Depth ◽  
Catalan Numbers ◽  
Hecke Groups ◽  
Polynomial Coefficients ◽  
Recursive Sequences ◽  
Counting Functions ◽  
Powers Of 2

We present a method to obtain congruences modulo powers of $2$ for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Fuß--Catalan numbers, and to subgroup counting functions associated with Hecke groups and their lifts. This leads to numerous new results, including many extensions of known results to higher powers of $2$.

scholarly journals Log-concavity of P-recursive sequences

2021 ◽  
Vol 107 ◽  
pp. 251-268
Author(s):  
Qing-hu Hou ◽  
Guojie Li
Keyword(s):  
Recursive Sequences

Aspects of Job Scheduling

1979 ◽  
Vol 101 (1) ◽  
pp. 17-22 ◽  
Author(s):  
K. Phillips
Keyword(s):  
Job Scheduling ◽  
Scheduling Problem ◽  
Deep Space Network ◽  
Fundamental Parameters ◽  
Space Network ◽  
Plant Operations ◽  
Mathematical Algorithms ◽  
Partitioning Problems ◽  
Combinatorial Methods ◽  
Operational Aspects

A mathematical model for job scheduling in a specified context is presented. The model uses both linear programming and combinatorial methods. While designed with a view toward optimization of scheduling of facility and plant operations at the Deep Space Network (DSN) Station at Goldstone, the context is sufficiently general to be widely applicable. The general scheduling problem including options for scheduling objectives is discussed and fundamental parameters identified. Mathematical algorithms for partitioning problems germaine to scheduling are presented. A more detailed description of algorithms and of operational aspects of the model is planned for a later report.

scholarly journals ON DIVISIBILITY OF BINOMIAL COEFFICIENTS

2012 ◽  
Vol 93 (1-2) ◽  
pp. 189-201 ◽  
Author(s):  
ZHI-WEI SUN
Keyword(s):  
Higher Order ◽  
Catalan Numbers ◽  
Binomial Coefficients

AbstractIn this paper, motivated by Catalan numbers and higher-order Catalan numbers, we study factors of products of at most two binomial coefficients.

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