Enumeration of k-Exceedance Lattice Paths with an Application to Comparing Chains of Order Statistics
Abstract
We enumerate the number of monotonic lattice paths starting at (0,0) and terminating at (m,n) in which l of the first k steps lie below the line y=x\ (0≤ k≤ m≤ n). These closed formulas consist of terms which are a product Catalan numbers, ballot numbers and binomial coefficients. We then apply the combinatorial formulas to failure analysis by deriving a probability distribution that compares the performance of a k-out-of-m system to a k-out-of-n system of continuous, independent, and identically distributed random variables. Lastly, we provide asymptotics in a few special cases of k,m,n and leave others as conjecture.
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