Journal of Mathematics and Statistics

Distribution of Records Defined on Ordered Words Representing Lattice Paths

A. M. Alahmadi and E. A. Mahmoud

DOI : 10.3844/jmssp.2011.184.186

Journal of Mathematics and Statistics

Volume 7, Issue 3

Pages 184-186

Abstract

Problem statement: Let the sequence i1,i2,…,in, denoted by Snn be an increasing ordered word of length n taken from the set of the n positive integers S= {1,2,…,n}, m, n ∈N+, m≥n.. Approach: That is 1≤i1≤i2≤…≤in≤n. Treating Snn as a sequence of weak records {Lj = ij}, i, j = 1,2,…,n, the distribution of the single weak record as well as the joint distribution of weak records were found before. Results: By defining the notion of strong records on the sequence {Lj = ij}, the distribution of a single strong record was found for m=n. In another aspect, it can be shown that the lattice path in the plane from (0,1) to (m, n) , consisting of unit segments up and to the right, can be represented by a sequence i1, i2,…, im where 1≤i1≤i2≤…≤im≤n, m, n ∈N+ m ≥ n. That is, such lattice paths can be represented, in one to one correspondence, by ordered increasing words of length m taken from the set S. Conclusion/Recommendations: In this article, we are going to extend the notion of weak and strong records to these sequences representing lattice paths for m>n and obtain their distributions. This result allows us to study lattice paths via ordered words of non negative integers.

Copyright

© 2011 A. M. Alahmadi and E. A. Mahmoud. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.