Open Journal of Modelling and Simulation
Vol.03 No.03(2015), Article ID:56936,6 pages
10.4236/ojmsi.2015.33007
Some New Results on the Number of Paths
Beih S. El-Desouky1, Abdelfattah Mustafa1, E. M. Mahmoud2
1Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt
2Mathematics Department, Faculty of Science, Aswan University, Aswan, Egypt
Email: b_desouky@yahoo.com, abdelfatah_mustafa@yahoo.com
Copyright © 2015 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
Received 23 April 2015; accepted 1 June 2015; published 5 June 2015
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 denoted by. 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
1. Introduction
Let be a sequence of natural numbers, and be an array associated with this sequence, whose entries such that
The path of order k along is defined to be a sequence of entries as follows
The number of paths of order k will be denoted by
By neglecting the last row in and then reconsidering it, we get the recurrence
(1)
When, a is a constant, then
(2)
and
Khidr and El-Desouky [1] proved that, when
(3)
where are the generalized Stirling numbers of the first kind associated with the sequence of real numbers, defined by [1] - [6] ,
(4)
These numbers satisfy the recurrence relation
(5)
And
Moreover, they introduced a special case of (3), when, then the number of paths of order k, is denoted by; and proved that
(6)
where are the Stirling numbers of the first kind defined by, see [2] [3]
Also the generating function for is given by
(7)
In this article, in Section 2, we derive a generalization of some results given in [1] , for the number of paths of
order k, , when. The generating function of is given. In Section 3, we find the probability distribution for and study some of their properties. The moment
generating function, skewness and kurtosis for are investigated. Moreover special case and numerical results are given in Section 4.
2. Main Results
Theorem 1. The number of paths of order k is given by
(8)
Proof. Using (5) in (8), we get
This by virtue of (1) completes the proof of (8).
Theorem 2. The generating function of the number of paths of order k is given by
(9)
Proof. Let the generating function of the number of paths of order k be denoted by
(10)
Using (1), we obtain
and hence we get
where. This completes the proof.
From (9), we get
where and hence we have
(11)
where
For the special case, we get
(12)
where
From (6) and (12), we have the identity
(13)
where
3. Some Applications
Let X, be the number of paths along, then by virtue of (8) we have
(14)
On the other hand the moment generating function of the random variable X denoted by, is given by the following theorem.
Theorem 3. The moment generating function of X, is given by
(15)
Proof. We begin by the definition of the moment generating function as follows.
This completes the proof.
Corollary 1. The jth moments of X is
(16)
Proof. The jth moments can be obtained from the moment generating function, where
This completes the proof.
Then from (16), we can calculate the mean and variance for the random variable X as follows.
(17)
(18)
hence the variance is given by
(19)
Corollary 2. The Skewness and kurtosis for the random variable X are given by
(20)
where
Proof. We can find the jth moments about the mean by using
(21)
From (16) and (21), we can find the moments about mean which can be used to calculate the skweness and kurtosis.
Special Case:
If, from (14), we have
and from (16) the jth moments has the form
and the mean is given by
the variance can be obtained as follows.
where we used, see [3] .
4. Numerical Results
Setting. Therefore the numerical values of, are reduced to, see [4] [5] .
From Equation (14), we can find the probability distribution of the number of paths X along as follows
From (16), we can compute the 4th moments as follows.
The 4th moments about mean can be obtained as
The values of mean and variance can be obtained from (17) and (19) as follows.
The skewness and kurtosis, respectively can be obtained from (20) as follows.
References
- 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
- Comtet, L. (1972) Nombres de Stirling generaux et fonctions symetriques. Comptes Rendus de l’Académie des Sci- ences Paris (Series A), 275, 747-750.
- Comtet, L. (1974) Advanced Combinatorics: The Art of Finite and Infinite Expansions. D. Reidel Publishing Company, Dordrecht, Holand.
- El-Desouky, B.S. (1994) Multiparameter Non-Central Stirling Numbers. The Fibonacci Quarterly, 32, 218-225.
- El-Desouky, B.S. and Cakić, 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
- Cakić, N.P., El-Desouky, B.S. and Milovanović, 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