Distribution of Records Defined on Ordered Words Representing Lattice Paths

Abstract

Problem statement: Let the sequence i₁,i₂,…,in, denoted by S_nⁿ 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≤…≤i_n≤n. Treating S_nⁿ as a sequence of weak records {L_j = i_j}, 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 {L_j = i_j}, 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,…, i_m where 1≤i₁≤i₂≤…≤i_m≤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.