# 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.