Some New Results on the Number of Paths
Affiliation(s)
1 Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt. 2 Mathematics Department, Faculty of Science, Aswan University, Aswan, Egypt.
1 Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt. 2 Mathematics Department, Faculty of Science, Aswan University, Aswan, Egypt.
ABSTRACT
Khidr and El-Desouky [1] derived a symmetric sum involving the Stirling numbers of the first kind through the process of counting the number of paths along a rectangular array n*m denoted by Anm. We investigate the generating function for the general case and hence some special cases as well. The probability function of the number of paths along is obtained. Moreover, the moment generating function of the random variable X and hence the mean and variance are obtained. Finally, some applications are introduced.
Khidr and El-Desouky [1] derived a symmetric sum involving the Stirling numbers of the first kind through the process of counting the number of paths along a rectangular array n*m denoted by Anm. We investigate the generating function for the general case and hence some special cases as well. The probability function of the number of paths along is obtained. Moreover, the moment generating function of the random variable X and hence the mean and variance are obtained. Finally, some applications are introduced.
KEYWORDS
Stirling Numbers, Generating Function, Moment Generating Function, Comtet Numbers, Maple Program
Stirling Numbers, Generating Function, Moment Generating Function, Comtet Numbers, Maple Program
Cite this paper
El-Desouky, B. , Mustafa, A. and Mahmoud, E. (2015) Some New Results on the Number of Paths. Open Journal of Modelling and Simulation, 3, 63-69. doi: 10.4236/ojmsi.2015.33007.
El-Desouky, B. , Mustafa, A. and Mahmoud, E. (2015) Some New Results on the Number of Paths. Open Journal of Modelling and Simulation, 3, 63-69. doi: 10.4236/ojmsi.2015.33007.
References
[1] Khidr, A.M. and El-Desouky, B.S. (1984) A Symmetric Sum Involving the Stirling Numbers of the First Kind. European Journal of Combinatorics, 5, 51-54.
http://dx.doi.org/10.1016/S0195-6698(84)80018-9 [2] Comtet, L. (1972) Nombres de Stirling generaux et fonctions symetriques. Comptes Rendus de l’Académie des Sciences Paris (Series A), 275, 747-750. [3] Comtet, L. (1974) Advanced Combinatorics: The Art of Finite and Infinite Expansions. D. Reidel Publishing Company, Dordrecht, Holand. [4] El-Desouky, B.S. (1994) Multiparameter Non-Central Stirling Numbers. The Fibonacci Quarterly, 32, 218-225. [5] El-Desouky, B.S. and Cakic, N.P. (2011) Generalized Higher Order Stirling Numbers. Mathematical and Computer Modelling, 54, 2848-2857.
http://dx.doi.org/10.1016/j.mcm.2011.07.005 [6] Cakic, N.P., El-Desouky, B.S. and Milovanovic, G.V. (2013) Explicit Formulas and Combinatorial Identities for Generalized Stirling Numbers. Mediterranean Journal of Mathematics, 10, 57-72.
http://dx.doi.org/10.1007/s00009-011-0169-x
[1] Khidr, A.M. and El-Desouky, B.S. (1984) A Symmetric Sum Involving the Stirling Numbers of the First Kind. European Journal of Combinatorics, 5, 51-54.
http://dx.doi.org/10.1016/S0195-6698(84)80018-9 [2] Comtet, L. (1972) Nombres de Stirling generaux et fonctions symetriques. Comptes Rendus de l’Académie des Sciences Paris (Series A), 275, 747-750. [3] Comtet, L. (1974) Advanced Combinatorics: The Art of Finite and Infinite Expansions. D. Reidel Publishing Company, Dordrecht, Holand. [4] El-Desouky, B.S. (1994) Multiparameter Non-Central Stirling Numbers. The Fibonacci Quarterly, 32, 218-225. [5] El-Desouky, B.S. and Cakic, N.P. (2011) Generalized Higher Order Stirling Numbers. Mathematical and Computer Modelling, 54, 2848-2857.
http://dx.doi.org/10.1016/j.mcm.2011.07.005 [6] Cakic, N.P., El-Desouky, B.S. and Milovanovic, G.V. (2013) Explicit Formulas and Combinatorial Identities for Generalized Stirling Numbers. Mediterranean Journal of Mathematics, 10, 57-72.
http://dx.doi.org/10.1007/s00009-011-0169-x